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

Answer 1

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

Given $f^{{}^{\prime }}\left(x\right)=\frac{1}{1+x^5}$
Since $g$ is inverse of $f$

$\therefore fog\left(x\right)=x$ for all $x$
$\frac{d}{dx}\left|fog\left(x\right)\right|=1$ for all $x$
$f^{{}^{\prime }}\left(g\left(x\right)\right).g^{{}^{\prime }}\left(x\right)=1$

$f^{{}^{\prime }}\left[g\left(x\right)\right]=\frac{1}{g^{{}^{\prime }}\left(x\right)}$

$\frac{1}{1+{\left[g\left(x\right)\right]}^5}=\frac{1}{g^{{}^{\prime }}\left(x\right)}$ ................................................$\because f^{{}^{\prime }}\left(x\right)=\frac{1}{1+x^5}$

$\therefore g^{{}^{\prime }}\left(x\right)=1+{\left[g\left(x\right)\right]}^5$