Question

Let $ g(x) $ be the inverse of an invertible function $ f(x) $ which is differentiable at $ x = c $ , then $ g'(f(c)) $ equals

• A. $ f'(c) $
• B. $ \frac{1}{f'(c)} $
• C. $ f(c) $
• D. $ \frac {1}{f(c)} $

Answer 1

Answer: (B)

Since, $g(x)$ is the inverse of an invertible function $f(x)$.
$\therefore g[f(x)]=x\,\,\,...(i)$
On differentiating Eq. (i) both sides w.r.t. ' $x$ ', we get
$g^{\prime}[f(x)] f^{\prime}(x)=1$
$\Rightarrow g^{\prime}[f(x)]=\frac{1}{f^{\prime}(x)} $
$\therefore g^{\prime}[f(c)]=\frac{1}{f^{\prime}(c)}$