How do you differentiate #f(x)=x^2(x+7)^3# using the product rule?

Answer 1

#f'(x)=(x+7)^2(x)(5x+14)#

#f(x)=(x)^2(x+7)^3#
Product rule: if #f(x) = AB#, then #f'(x)=AB' + A'B#

In this instance, the chain rule must be applied in order to distinguish the terms "A" and "B".

#f'(x)=(x)^2(3)(x+7)^2+(2x)(x+7)^3#
Now simplify by factoring: #f'(x)=(x+7)^2(x)[3x+2(x+7)]# #f'(x)=(x+7)^2(x)(3x+2x+14)# #f'(x)=(x+7)^2(x)(5x+14)#
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Answer 2

# :. f'(x) = x(x+7)^2 (5x+14} #

The Product Rule for Differentiation is something you should learn and practice using if you are studying math:

# d/dx(uv)=u(dv)/dx+v(du)/dx #, or, # (uv)' = (du)v + u(dv) #

"The first times the derivative of the second plus the second times the derivative of the first" is the rule that I was taught to memorize.

So with # f(x)=x^2(x+7)^3 # Then
# { ("Let "u=x^2, => , (du)/dx=2x), ("And "v=(x+7)^3, =>, (dv)/dx=3(x+7)^2 " (by chain rule)" ) :}#
# :. d/dx(uv) = u(dv)/dx+v(du)/dx # # :. f'(x) = (x^2)(3(x+7)^2) + ((x+7)^3)(2x) # # :. f'(x) = 3x^2(x+7)^2 + 2x(x+7)^3 # # :. f'(x) = x(x+7)^2 {3x+2(x+7)} # # :. f'(x) = x(x+7)^2 (3x+2x+14} # # :. f'(x) = x(x+7)^2 (5x+14} #
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Answer 3

To differentiate ( f(x) = x^2(x+7)^3 ) using the product rule, you can follow these steps:

  1. Identify the functions ( u ) and ( v ) where ( f(x) = u(x) \cdot v(x) ). Let ( u(x) = x^2 ) and ( v(x) = (x+7)^3 ).

  2. Apply the product rule, which states that the derivative of the product of two functions ( u ) and ( v ) is given by: [ \frac{d}{dx}[u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x) ]

  3. Differentiate each function separately: [ u'(x) = 2x ] [ v'(x) = 3(x+7)^2 ]

  4. Substitute the derivatives and the original functions into the product rule formula: [ f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x) ] [ f'(x) = (2x) \cdot (x+7)^3 + (x^2) \cdot 3(x+7)^2 ]

  5. Simplify the expression: [ f'(x) = 2x(x+7)^3 + 3x^2(x+7)^2 ]

This is the derivative of the function ( f(x) = x^2(x+7)^3 ) using the product rule.

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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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