2 like 0 dislike
1,226 views

Based on the table of values for the differentiable, invertible function $f$ and its derivative, evaluate $\left(f^{-1}\right)^{\prime}(2)$.

reopened | 1,226 views

0 like 0 dislike

$\left(f^{-1}\right)^{\prime}(2)=-\frac{1}{4}$

Explanation:

We know that
$f\left(\left(f^{-1}\right)(x)\right)=x$
Differentiating both sides with respect to $x$ we get
\begin{aligned} &\mathrm{f}^{\prime}\left(\left(\mathrm{f}^{-1}\right)(\mathrm{x})\right) \cdot\left(\left(\mathrm{f}^{-1}\right)^{\prime}(\mathrm{x})\right)=1 \\ &\Longrightarrow\left(\mathrm{f}^{-1}\right)^{\prime}(\mathrm{x})=\frac{1}{\mathrm{f}^{\prime}\left(\left(\mathrm{f}^{-1}\right)(\mathrm{x})\right)} \end{aligned}
Putting $x=2$ we get
\begin{aligned} &\Rightarrow\left(f^{-1}\right)^{\prime}(2)=\frac{1}{f^{\prime}\left(\left(f^{-1}\right)(2)\right)} \\ &\Longrightarrow\left(f^{-1}\right)^{\prime}(2)=\frac{1}{f^{\prime}(3)}(\because f(3)=2) \\ &\Longrightarrow\left(f^{-1}\right)^{\prime}(2)=\frac{1}{-4} \\ &\Longrightarrow\left(f^{-1}\right)^{\prime}(2)=-\frac{1}{4} \end{aligned}

by Gold Status (31,743 points)

1 like 0 dislike
0 like 0 dislike
2 like 0 dislike
1 like 0 dislike
0 like 0 dislike
1 like 0 dislike
1 like 0 dislike
0 like 0 dislike
0 like 0 dislike