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Derivative of an Inverse Function
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Rank the following steps for finding the derivative of an inverse function, $$\left(f^{-1}\right)'(a)$$, given a function $$f(x)$$ (from first to last):

A) Identify the $$x$$-value such that $$f(x)=a$$.
B) Compute the derivative $$f'(x)$$ at that $$x$$.
C) Take the reciprocal of $$f'(x)$$, which yields $$\left(f^{-1}\right)'(a)$$.
D) Simplify the expression to obtain the final answer.

A

Identify the x-value where $$f(x)=a$$, compute $$f'(x)$$ at that point, take the reciprocal to obtain $$\left(f^{-1}\right)'(a)$$, and simplify the result

B

Compute $$f'(a)$$ directly, take its reciprocal, substitute back, and simplify

C

Identify the y-value corresponding to $$a$$, differentiate $$f^{-1}(y)$$ with respect to $$y$$, and then substitute $$y=a$$

D

Differentiate the inverse function directly using the chain rule, then compute $$f'(a)$$, and take the reciprocal

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