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How do you differentiate #f(x) = (3x+7)^5?# using the chain rule?

Answer 1

Evelyn Agard

#f'(x)=15(3x+7)^4#

#f(x)= (3x+7)^5#

Chain rule: #f(g(x))= f'(g(x))*g'(x)#

#f'(x)= 5(3x+7)^(5-1)*d/dx(3x+7)#

#=5(3x+7)^4 (3)# #=15(3x+7)^4#

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Answer 2

Emilia Aguilar

To differentiate ( f(x) = (3x+7)^5 ) using the chain rule, you first identify the inner function and its derivative, then apply the chain rule formula which states that the derivative of a composite function ( f(g(x)) ) is ( f'(g(x)) \cdot g'(x) ).

Let ( u = 3x + 7 ). Then ( f(x) = u^5 ). The derivative of ( u ) with respect to ( x ) is ( \frac{du}{dx} = 3 ).

Now, using the chain rule, ( \frac{df}{dx} = \frac{df}{du} \cdot \frac{du}{dx} ).

The derivative of ( u^5 ) with respect to ( u ) is ( 5u^4 ).

Therefore, ( \frac{df}{dx} = 5u^4 \cdot 3 = 15(3x + 7)^4 ).

So, ( \frac{df}{dx} = 15(3x + 7)^4 ) is the derivative of ( f(x) = (3x+7)^5 ) using the chain 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.