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,
$f(g(x))=x$ (Property of inverse function)
Differentiating both sides

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

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 $f(p)=2$

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

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