Question

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

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

Answer 1

The correct option is B

${1+\left\{g\right(x\left)\right\}}^5$

Since $f\left(x\right)$ and $g\left(x\right)$ are inverse of each other
$\therefore g\left(f\right(x\left)\right)=x\Rightarrow g^{\prime }\left(f\right(x\left)\right)f^{\prime }\left(x\right)=1\Rightarrow g^{\prime }\left(f\right(x\left)\right)=1+x^5(\because f^{\prime }(x)=\frac{1}{1+x^5})$
Here$f\left(x\right)=y⟹x=g\left(y\right)$
$\therefore g^{\prime }\left(y\right)=1+\left\{g\right(y){\}}^5\Rightarrow g^{\prime }(x)=1+\{g\left(x\right){\}}^5$