How do you differentiate #f(x)=x^2 (4x^3 - 1) # using the product rule?

Answer 1

#f'(x)=x(11x^3-2)#

In line with the product rule:

#f'(x)=(4x^3-1)color(blue)(d/dx[x^2])+x^2color(red)(d/dx[4x^3-1])#

Determine every derivative.

#f'(x)=2x(4x^3-1)+x^2(3x^2)#
#f'(x)=8x^4-2x+3x^4#
#f'(x)=x(11x^3-2)#
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Answer 2

To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formulaTo differentiate the function ( f(x) = x^2(4x^3To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:To differentiate the function ( f(x) = x^2(4x^3 - To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

To differentiate the function ( f(x) = x^2(4x^3 - 1To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

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[ \frac{To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule,To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{dTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the twoTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dxTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functionsTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx}To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions beingTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multipliedTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [uTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied:To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(xTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: (To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)]To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( uTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = uTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = xTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(xTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 \To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x)To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 )To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(xTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) andTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x)To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and (To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) +To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( vTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + uTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(xTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(xTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x)To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x)To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) =To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) vTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(xTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4xTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x)To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. ApplyTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( vTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: (To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(xTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x)To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uvTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)'To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' =To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4xTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v +To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ). To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ). 3.To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 -To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. FindTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find theTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivativesTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 \To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives ofTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of (To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

1To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( uTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

1.To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(xTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. FindTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  2. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  3. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  4. Find the derivatives of ( u(x)To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find (To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  2. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  3. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  4. Find the derivatives of ( u(x) \To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( uTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  2. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  3. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  4. Find the derivatives of ( u(x) )To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  2. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  3. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  4. Find the derivatives of ( u(x) ) andTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(xTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  2. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  3. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  4. Find the derivatives of ( u(x) ) and (To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x)To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  2. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  3. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  4. Find the derivatives of ( u(x) ) and ( vTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) \To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  2. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  3. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  4. Find the derivatives of ( u(x) ) and ( v(xTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ),To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  2. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  3. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  4. Find the derivatives of ( u(x) ) and ( v(x)To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), theTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  2. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  3. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  4. Find the derivatives of ( u(x) ) and ( v(x) \To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivativeTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  2. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  3. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  4. Find the derivatives of ( u(x) ) and ( v(x) ): To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative ofTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  2. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  3. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  4. Find the derivatives of ( u(x) ) and ( v(x) ): To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of (To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  2. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  3. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  4. Find the derivatives of ( u(x) ) and ( v(x) ): -To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(xTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  2. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  3. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  4. Find the derivatives of ( u(x) ) and ( v(x) ):

    • (To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x)To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  2. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  3. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  4. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) \To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  2. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  3. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  4. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(xTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ),To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  2. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  3. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  4. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x)To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), whichTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  2. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  3. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  4. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) =To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which isTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  2. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  3. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  4. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is (To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  2. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  3. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  4. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  2. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  3. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  4. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x \To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2xTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  2. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  3. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  4. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x )To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x \To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  2. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  3. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  4. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ). To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  2. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  3. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  4. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (derTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ). 2To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  2. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  3. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  4. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (derivativeTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).

  2. FindTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  3. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  4. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  5. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (derivative ofTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).

  2. Find (To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  3. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  4. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  5. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (derivative of (To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).

  2. Find ( vTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  3. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  4. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  5. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (derivative of ( x^To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).

  2. Find ( v'(xTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  3. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  4. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  5. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (derivative of ( x^2To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).

  2. Find ( v'(x)To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  3. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  4. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  5. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (derivative of ( x^2 \To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).

  2. Find ( v'(x) \To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  3. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  4. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  5. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (derivative of ( x^2 )). To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).

  2. Find ( v'(x) ),To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  3. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  4. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  5. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (derivative of ( x^2 )). To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).

  2. Find ( v'(x) ), theTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  3. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  4. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  5. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (derivative of ( x^2 )). -To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).

  2. Find ( v'(x) ), the derivativeTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  3. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  4. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  5. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • (To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).

  2. Find ( v'(x) ), the derivative ofTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  3. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  4. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  5. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( vTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).

  2. Find ( v'(x) ), the derivative of (To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  3. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  4. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  5. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).

  2. Find ( v'(x) ), the derivative of ( vTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  3. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  4. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  5. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(xTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).

  2. Find ( v'(x) ), the derivative of ( v(xTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  3. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  4. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  5. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x)To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).

  2. Find ( v'(x) ), the derivative of ( v(x)To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  3. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  4. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  5. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) =To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).

  2. Find ( v'(x) ), the derivative of ( v(x) \To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  3. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  4. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  5. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).

  2. Find ( v'(x) ), the derivative of ( v(x) ), whichTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  3. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  4. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  5. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).

  2. Find ( v'(x) ), the derivative of ( v(x) ), which isTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  3. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  4. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  5. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).

  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  3. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  4. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  5. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).

  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  3. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  4. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  5. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 \To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).

  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12xTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  3. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  4. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  5. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 )To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).

  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  3. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  4. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  5. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).

  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  3. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  4. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  5. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).

  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 \To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  3. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  4. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  5. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  6. Substitute these derivativesTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).

  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ). To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  3. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  4. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  5. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  6. Substitute these derivatives intoTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).

  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ). 3To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  3. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  4. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  5. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  6. Substitute these derivatives into the productTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).

  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ). 3.To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  3. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  4. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  5. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  6. Substitute these derivatives into the product ruleTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).

  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).

  3. ApplyTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  4. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  5. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  6. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  7. Substitute these derivatives into the product rule formulaTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).

  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).

  3. Apply theTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  4. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  5. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  6. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  7. Substitute these derivatives into the product rule formula: To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).

  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).

  3. Apply the productTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  4. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  5. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  6. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  7. Substitute these derivatives into the product rule formula: To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).

  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).

  3. Apply the product ruleTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  4. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  5. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  6. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  7. Substitute these derivatives into the product rule formula: -To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).

  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).

  3. Apply the product rule:To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  4. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  5. Apply the product rule formula: ( (uv)' = u'v + uv' ).

  6. Find the derivatives of ( u(x) ) and ( v(x) ):

    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  7. Substitute these derivatives into the product rule formula:

    • (To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (xTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ fTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x)To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) vTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) +To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4xTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + uTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(xTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 -To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x)To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(xTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x)To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 -To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) +To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 -To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + xTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1)To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (xTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2)To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) \To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ). To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ). 5To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12xTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ). 5.To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. SimplTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. SimplifyTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2)To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expressionTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2) ]

[To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expression: To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2) ]

[ fTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expression: To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2) ]

[ f'(To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expression:
    • (To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2) ]

[ f'(xTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expression:
    • ( (x^2(4xTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2) ]

[ f'(x) = 8x^4To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expression:
    • ( (x^2(4x^To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2) ]

[ f'(x) = 8x^4 -To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expression:
    • ( (x^2(4x^3 -To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2) ]

[ f'(x) = 8x^4 - To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expression:
    • ( (x^2(4x^3 - To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2) ]

[ f'(x) = 8x^4 - 2To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expression:
    • ( (x^2(4x^3 - 1To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2) ]

[ f'(x) = 8x^4 - 2xTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expression:
    • ( (x^2(4x^3 - 1))To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2) ]

[ f'(x) = 8x^4 - 2x +To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expression:
    • ( (x^2(4x^3 - 1))'To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2) ]

[ f'(x) = 8x^4 - 2x + To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expression:
    • ( (x^2(4x^3 - 1))' =To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2) ]

[ f'(x) = 8x^4 - 2x + 12To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expression:
    • ( (x^2(4x^3 - 1))' = To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2) ]

[ f'(x) = 8x^4 - 2x + 12xTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expression:
    • ( (x^2(4x^3 - 1))' = 8xTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2) ]

[ f'(x) = 8x^4 - 2x + 12x^To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expression:
    • ( (x^2(4x^3 - 1))' = 8x^To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2) ]

[ f'(x) = 8x^4 - 2x + 12x^4To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expression:
    • ( (x^2(4x^3 - 1))' = 8x^4To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2) ]

[ f'(x) = 8x^4 - 2x + 12x^4 ]

To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expression:
    • ( (x^2(4x^3 - 1))' = 8x^4 -To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2) ]

[ f'(x) = 8x^4 - 2x + 12x^4 ]

[To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expression:
    • ( (x^2(4x^3 - 1))' = 8x^4 - 2To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2) ]

[ f'(x) = 8x^4 - 2x + 12x^4 ]

[ fTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expression:
    • ( (x^2(4x^3 - 1))' = 8x^4 - 2xTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2) ]

[ f'(x) = 8x^4 - 2x + 12x^4 ]

[ f'(xTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expression:
    • ( (x^2(4x^3 - 1))' = 8x^4 - 2x +To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2) ]

[ f'(x) = 8x^4 - 2x + 12x^4 ]

[ f'(x)To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expression:
    • ( (x^2(4x^3 - 1))' = 8x^4 - 2x + 12To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2) ]

[ f'(x) = 8x^4 - 2x + 12x^4 ]

[ f'(x) =To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expression:
    • ( (x^2(4x^3 - 1))' = 8x^4 - 2x + 12xTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2) ]

[ f'(x) = 8x^4 - 2x + 12x^4 ]

[ f'(x) = To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expression:
    • ( (x^2(4x^3 - 1))' = 8x^4 - 2x + 12x^To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2) ]

[ f'(x) = 8x^4 - 2x + 12x^4 ]

[ f'(x) = 20To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expression:
    • ( (x^2(4x^3 - 1))' = 8x^4 - 2x + 12x^4To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2) ]

[ f'(x) = 8x^4 - 2x + 12x^4 ]

[ f'(x) = 20xTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expression:
    • ( (x^2(4x^3 - 1))' = 8x^4 - 2x + 12x^4 \To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2) ]

[ f'(x) = 8x^4 - 2x + 12x^4 ]

[ f'(x) = 20x^To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expression:
    • ( (x^2(4x^3 - 1))' = 8x^4 - 2x + 12x^4 ). To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2) ]

[ f'(x) = 8x^4 - 2x + 12x^4 ]

[ f'(x) = 20x^4To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expression:
    • ( (x^2(4x^3 - 1))' = 8x^4 - 2x + 12x^4 ). 6To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2) ]

[ f'(x) = 8x^4 - 2x + 12x^4 ]

[ f'(x) = 20x^4 -To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expression:
    • ( (x^2(4x^3 - 1))' = 8x^4 - 2x + 12x^4 ). 6.To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2) ]

[ f'(x) = 8x^4 - 2x + 12x^4 ]

[ f'(x) = 20x^4 - To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expression:
    • ( (x^2(4x^3 - 1))' = 8x^4 - 2x + 12x^4 ).
  6. CombineTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2) ]

[ f'(x) = 8x^4 - 2x + 12x^4 ]

[ f'(x) = 20x^4 - 2To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expression:
    • ( (x^2(4x^3 - 1))' = 8x^4 - 2x + 12x^4 ).
  6. Combine likeTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2) ]

[ f'(x) = 8x^4 - 2x + 12x^4 ]

[ f'(x) = 20x^4 - 2xTo differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expression:
    • ( (x^2(4x^3 - 1))' = 8x^4 - 2x + 12x^4 ).
  6. Combine like termsTo differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2) ]

[ f'(x) = 8x^4 - 2x + 12x^4 ]

[ f'(x) = 20x^4 - 2x \To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expression:
    • ( (x^2(4x^3 - 1))' = 8x^4 - 2x + 12x^4 ).
  6. Combine like terms: To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2) ]

[ f'(x) = 8x^4 - 2x + 12x^4 ]

[ f'(x) = 20x^4 - 2x ]To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expression:
    • ( (x^2(4x^3 - 1))' = 8x^4 - 2x + 12x^4 ).
  6. Combine like terms: To differentiate ( f(x) = x^2 (4x^3 - 1) ) using the product rule, you apply the formula:

[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) ]

where ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).

  1. Find ( u'(x) ), the derivative of ( u(x) ), which is ( 2x ).
  2. Find ( v'(x) ), the derivative of ( v(x) ), which is ( 12x^2 ).
  3. Apply the product rule:

[ f'(x) = u'(x) v(x) + u(x) v'(x) ]

[ f'(x) = (2x)(4x^3 - 1) + (x^2)(12x^2) ]

[ f'(x) = 8x^4 - 2x + 12x^4 ]

[ f'(x) = 20x^4 - 2x ]To differentiate the function ( f(x) = x^2(4x^3 - 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = x^2 ) and ( v(x) = 4x^3 - 1 ).
  2. Apply the product rule formula: ( (uv)' = u'v + uv' ).
  3. Find the derivatives of ( u(x) ) and ( v(x) ):
    • ( u'(x) = 2x ) (derivative of ( x^2 )).
    • ( v'(x) = 12x^2 ) (derivative of ( 4x^3 - 1 )).
  4. Substitute these derivatives into the product rule formula:
    • ( (x^2(4x^3 - 1))' = (2x)(4x^3 - 1) + x^2(12x^2) ).
  5. Simplify the expression:
    • ( (x^2(4x^3 - 1))' = 8x^4 - 2x + 12x^4 ).
  6. Combine like terms:
    • ( (x^2(4x^3 - 1))' = 20x^4 - 2x ).

Therefore, the derivative of ( f(x) = x^2(4x^3 - 1) ) using the product rule is ( f'(x) = 20x^4 - 2x ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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