Question

Let the functions $f\left(x\right)$ and $g\left(x\right)$ be inverse function of each other. If both th functions are differentiable at $x=c$, then $g^{\prime }\left(f\right(c\left)\right)$ =

• A. $f^{\prime }\left(c\right)$
• B. $-f^{\prime }\left(c\right)$
• C. $\frac{1}{f^{\prime }\left(c\right)}$
• D. $-\frac{1}{f^{\prime }\left(c\right)}$

Answer 1

The correct option is C $\frac{1}{f^{\prime }\left(c\right)}$

We know that $f\left(x\right)$ and $g\left(x\right)$ are inverse function of each other, so
$g\left(f\right(x\left)\right)=x$
Differentiating w.r.t. $x$,
$\Rightarrow g^{\prime }\left(f\right(x\left)\right).f^{\prime }\left(x\right)=1$
$\therefore g^{\prime }\left(f\right(c\left)\right)$ = $\frac{1}{f^{\prime }\left(c\right)}$