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

$\begin{array}{c}gisinverseoffunctionf,then\\ f\left(g\left(x\right)\right)=x\\ differentiatingwithrespecttox\\ f^{{}^{\prime }}\left(g\left(x\right)\right)\times g^{{}^{\prime }}\left(x\right)=1\\ so{g}^{{}^{\prime }}\left(x\right)=\frac{1}{f^{{}^{\prime }}\left(g\left(x\right)\right)}\\ =1+{\left\{g\left(x\right)\right\}}^5\end{array}$