Question

Let $f$ and $g$ be differentiable function satisfying $g^{\prime }\left(a\right)=2,g\left(a\right)=b$ and $g$ is the inverse of $f$. Then $f^{\prime }\left(b\right)$ is equal to

• A. $\frac{1}{2}$
• B. $1$
• C. $2$
• D. $\frac{2}{3}$

Answer 1

The correct option is A $\frac{1}{2}$

Given : $g$ is the inverse of $f$
$\Rightarrow f^{-1}\left(x\right)=g\left(x\right)$
$\Rightarrow f\left(g\right(x\left)\right)=x$
Differentiating both sides w.r.t. $x$
$f^{\prime }\left(g\right(x\left)\right)\cdot g^{\prime }\left(x\right)=1$
$\Rightarrow f^{\prime }\left(g\right(x\left)\right)=\frac{1}{g^{\prime }\left(x\right)}$
Put $x=a$
$\Rightarrow f^{\prime }\left(g\right(a\left)\right)=\frac{1}{g^{\prime }\left(a\right)}$
$\therefore f^{\prime }\left(b\right)=\frac{1}{2}$