Differentiation of Inverse Function - Practice Questions & MCQ

Differentiation of Inverse Function

Let $f(x)$ be a function that is both invertible and differentiable. Let $y=g(x)$ be the inverse of $f(x)$. Then,
$f(g(x))=x \quad$ (Property of inverse function)
Differentiating both sides

$
\begin{aligned}
& f^{\prime}(g(x)) \cdot g^{\prime}(x)=1 \quad \text { (Chain Rule) } \\
& g^{\prime}(x)=\frac{1}{f^{\prime}(g(x))}
\end{aligned}
$

Illustration

If $f(x)=x^3+x^5$, and $g(x)$ is the inverse of $f(x)$, then find $g^{\prime}(2)$
Solution
Using $g^{\prime}(x)=\frac{1}{f^{\prime}(g(x))}$, put $\mathrm{x}=2$

$
g^{\prime}(2)=\frac{1}{f^{\prime}(g(2))}
$
Now we need to get the value of $g(2)$
As we know for inverse functions if $f(a)=b$, then $g(b)=a$. So let $g(2)=p$, then $f(p)=2$

$
\begin{aligned}
& p^3+p^5=2 \\
& p=1=g(2)
\end{aligned}
$
Putting this in (i)

$
g^{\prime}(2)=\frac{1}{f^{\prime}(1)}=\frac{1}{f^{\prime}(1)}=\frac{1}{3 x^2+5 x^4}(\text { at } x=1)=\frac{1}{8}
$