How do you differentiate #f(x)= (2x^2-5)(x+1) # using the product rule?

Answer 1

#f'(x)=6x^2+4x-5#

The product rule is #k'(x)=f(x)g'(x)+g(x)f'(x)#

Written informally, this is:

(derivative of product)=(first)x(derivative of second)+(second)x(derivative of the first).

In this case the 'first' is #(2x^2-5)# and the 'second' is #(x+1)#.
The derivative of the 'first' is calculated as follows: #d/dx(2x^2-5)=d/dx2x^2+d/dx-5#
#=2d/dxx^2-d/dx5#
#=2*2x-0#
#=4x#

The derivative of the 'second' is calculated:

#d/dx(x+1)=d/dxx+d/dx1#
#=1+0=1#

Using the product rule (and the derivatives worked out above) gives:

#f'(x)=(2x^2-5)(1)+(x+1)(4x)#
#=2x^2-5+4x^2+4x#
#=6x^2+4x-5#
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Answer 2

To differentiate ( f(x) = (2x^2 - 5)(x + 1) ) using the product rule, follow these steps:

  1. Identify the two functions being multiplied: ( u(x) = 2x^2 - 5 ) and ( v(x) = x + 1 ).
  2. Apply the product rule formula: ( f'(x) = u'(x)v(x) + u(x)v'(x) ).
  3. Differentiate ( u(x) ) and ( v(x) ) separately:
    • ( u'(x) = \frac{d}{dx}(2x^2 - 5) = 4x )
    • ( v'(x) = \frac{d}{dx}(x + 1) = 1 ).
  4. Substitute the derivatives and the original functions into the product rule formula:
    • ( f'(x) = (4x)(x + 1) + (2x^2 - 5)(1) ).
  5. Simplify the expression:
    • ( f'(x) = 4x^2 + 4x + 2x^2 - 5 ).
  6. Combine like terms:
    • ( f'(x) = 6x^2 + 4x - 5 ).

So, the derivative of ( f(x) ) with respect to ( x ) using the product rule is ( 6x^2 + 4x - 5 ).

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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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