Differentiating Inverse Functions Practice

Sample questions

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1. Sample 1difficulty 3/5

   Suppose $f(1) = 0$, $f'(1) = 5$. Then $(f^{-1})'(0) =$

   • A

     $0$

   • B

     $5$

   • C

     Undefined

   • D

     $\tfrac{1}{5}$

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   Why

   $(f^{-1})'(0) = 1/f'(f^{-1}(0)) = 1/f'(1) = 1/5$.

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2. Sample 2difficulty 3/5

   If $f$ has inverse $f^{-1}$ and $f(2) = 5$, $f'(2) = 4$, then $(f^{-1})'(5) =$

   • A

     $\tfrac{1}{4}$

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   • B

     $4$

   • C

     $\tfrac{1}{5}$

   • D

     $5$

   Why

   $(f^{-1})'(b) = 1/f'(a)$ where $f(a) = b$. Here $1/f'(2) = 1/4$.

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3. Sample 3difficulty 4/5

   If $f(x) = x^3 + x$ and $g$ is its inverse, then $g'(2)$ equals

   • A

     $\frac{1}{2}$

   • B

     $4$

   • C

     $\frac{1}{4}$

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   • D

     $2$

   Why

   $f(1) = 2$, so $g(2)=1$. $g'(2) = 1/f'(1) = 1/(3+1) = 1/4$.

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