2.8 Derivatives of an Inverse Function
The Derivative of an Inverse Function
Let $f$ be a differentiable function whose inverse function $f^{-1}$ is also differentiable. Then, providing that the denominator is not zero,
$
\left(f^{-1}\right)^{\prime}(x)=\frac{1}{f^{\prime}\left(f^{-1}(x)\right)} \quad \text { or } \quad\left(f^{-1}\right)^{\prime}(a)=\frac{1}{f^{\prime}\left(f^{-1}(a)\right)}
$
Example
- Let $f(x)=x^2-\frac{3}{x}$.
(a) What is the value of $f^{-1}(8)$ ?
(b) What is the value of $\left(f^{-1}\right)^{\prime}(8)$ ?
▶️Answer/Explanation
Solution
(a) $f(x)=x^2-\frac{3}{x}=8$, when $x=3$.
Since $f(3)=8, f^{-1}(8)=3$.
$
\begin{aligned}
& f^{\prime}(x)=2 x-3(-1) x^{-2}=2 x+\frac{3}{x^2} \\
& f^{\prime}(3)=2(3)+\frac{3}{3^2}=\frac{19}{3} \\
& \left(f^{-1}\right)^{\prime}(8)=\frac{1}{f^{\prime}\left(f^{-1}(8)\right)}=\frac{1}{f^{\prime}(3)} \quad f^{-1}(8)=3 \\
& =\frac{1}{19 / 3}=\frac{3}{19}
\end{aligned}
$
2.8 Derivatives of an Inverse Function
The Derivative of an Inverse Function
Let $f$ be a differentiable function whose inverse function $f^{-1}$ is also differentiable. Then, providing that the denominator is not zero,
$
\left(f^{-1}\right)^{\prime}(x)=\frac{1}{f^{\prime}\left(f^{-1}(x)\right)} \quad \text { or } \quad\left(f^{-1}\right)^{\prime}(a)=\frac{1}{f^{\prime}\left(f^{-1}(a)\right)}
$
Example
- Let $f(x)=x^2-\frac{3}{x}$.
(a) What is the value of $f^{-1}(8)$ ?
(b) What is the value of $\left(f^{-1}\right)^{\prime}(8)$ ?
▶️Answer/Explanation
Solution
(a) $f(x)=x^2-\frac{3}{x}=8$, when $x=3$.
Since $f(3)=8, f^{-1}(8)=3$.
$
\begin{aligned}
& f^{\prime}(x)=2 x-3(-1) x^{-2}=2 x+\frac{3}{x^2} \\
& f^{\prime}(3)=2(3)+\frac{3}{3^2}=\frac{19}{3} \\
& \left(f^{-1}\right)^{\prime}(8)=\frac{1}{f^{\prime}\left(f^{-1}(8)\right)}=\frac{1}{f^{\prime}(3)} \quad f^{-1}(8)=3 \\
& =\frac{1}{19 / 3}=\frac{3}{19}
\end{aligned}
$
Example
- If $f(2)=5$ and $f^{\prime}(2)=\frac{1}{4}$, find $\left(f^{-1}\right)^{\prime}(5)$.
▶️Answer/Explanation
Solution
$
\begin{aligned}
& \text { Since } f(2)=5, f^{-1}(5)=2 \\
& \left(f^{-1}\right)^{\prime}(5)=\frac{1}{f^{\prime}\left(f^{-1}(5)\right)}=\frac{1}{f^{\prime}(2)}=\frac{1}{1 / 4}=4
\end{aligned}
$
Exercises – Derivatives of an Inverse Function
Multiple Choice Questions
- 1. Let $f$ and $g$ be functions that are differentiable everywhere. If $g$ is the inverse function of $f$ and if $g(3)=4$ and $f^{\prime}(4)=\frac{3}{2}$, then $g^{\prime}(3)=$
(A) $\frac{1}{4}$
(B) $\frac{1}{3}$
(C) $\frac{2}{3}$
(D) $\frac{4}{3}$
▶️Answer/Explanation
Ans:C
Example
- If $f(-3)=2$ and $f^{\prime}(-3)=\frac{3}{4}$, then $\left(f^{-1}\right)^{\prime}(2)=$
(A) $\frac{1}{2}$
(B) $\frac{4}{3}$
(C) $\frac{3}{2}$
(D) $-\frac{3}{4}$
▶️Answer/Explanation
Ans:B
Example
- If $f(x)=x^3-x+2$, then $\left(f^{-1}\right)^{\prime}(2)=$
(A) $\frac{1}{2}$
(B) $\frac{2}{3}$
(C) 4
(D) 6
▶️Answer/Explanation
Ans:A
Example
- If $f(x)=x^3-x+2$, then $\left(f^{-1}\right)^{\prime}(2)=$
(A) $\frac{1}{2}$
(B) $\frac{2}{3}$
(C) 4
(D) 6
▶️Answer/Explanation
Ans:D
Example
- 4. If $f(x)=\sin x$, then $\left(f^{-1}\right)^{\prime}\left(\frac{\sqrt{3}}{2}\right)=$
(A) $\frac{1}{2}$
(B) $\frac{2 \sqrt{3}}{3}$
(C) $\sqrt{3}$
(D) 2
▶️Answer/Explanation
Ans:D