Question

If $g$ is the inverse of a function $f$ and $f\text{'}\left(x\right)=\frac{1}{1+x^5}$, then $g\text{'}\left(x\right)$ is equal to:

• A. $1+x^5$
• B. $5x^4$
• C. $\frac{1}{1+\left\{g\right(x){\}}^5}$
• D. $1+\left\{g\right(x){\}}^5$

Answer 1

The correct option is D $1+\left\{g\right(x){\}}^5$

$g\left(x\right)=f^{-1}x$
$\Rightarrow f\left(g\right(x\left)\right)=x$
$\Rightarrow f^{\prime }\left(g\right(x\left)\right)\cdot g^{\prime }\left(x\right)=1$
$\Rightarrow g^{\prime }\left(x\right)=\frac{1}{f^{\prime }\left(g\right(x\left)\right)}...\left(1\right)$
As given, $f^{\prime }\left(x\right)=\frac{1}{1+x^5}$
$\Rightarrow f^{\prime }\left(g\right(x\left)\right)=\frac{1}{1+\left\{g\right(x){\}}^5}$
from equation (1)
$g^{\prime }\left(x\right)=1+\left\{g\right(x){\}}^5$