Mathematics - Functions Question with Solution | TestHub

$g$ is the inverse of $f$ i.e.  $g\left(x\right)=f^{-1}\left(x\right)$

$\Rightarrow f\left(g\left(x\right)\right)=x$

Differentiating both the sides w.r.t $x$ , we get

$f^{\text{'}}\left(g\left(x\right)\right).g^{\text{'}}\left(x\right)=1$

$\Rightarrow g^{\text{'}}\left(x\right)=\frac{1}{f^{\text{'}}\left(g\left(x\right)\right)}$as $\left( f^{\text{'}}\left(x\right)=\frac{1}{1+x^5}\right)$

$\Rightarrow g^{\text{'}}\left(x\right)=\frac{1}{\left(\frac{1}{1+{\left(g\left(x\right)\right)}^5}\right)}$

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