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How do you differentiate #f(x) = (2x+1)^7# using the chain rule?

Answer 1

Isabella Ackerman

You can do it like this:

#f(x)=(2x+1)^7#

Treat #(2x+1)# as it were a single function. Differentiate that and then multiply by the differential of #(2x+1)rArr#

#f'(x)=7(2x+1)^(6)xx2#

#f'(x)=14(2x+1)^6#

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Answer 2

Emily Ammerman

To differentiate ( f(x) = (2x+1)^7 ) using the chain rule, you first identify the outer function as ( u^7 ) and the inner function as ( u = 2x + 1 ). Then, you apply the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

So, the derivative of ( f(x) ) with respect to ( x ) is:

[ \frac{d}{dx}[(2x+1)^7] = 7(2x+1)^{7-1} \cdot \frac{d}{dx}(2x+1) ]

[ = 7(2x+1)^6 \cdot 2 ]

[ = 14(2x+1)^6 ]

Therefore, the derivative of ( f(x) = (2x+1)^7 ) using the chain rule is ( f'(x) = 14(2x+1)^6 ).

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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.