Question

If $g$ is the inverse of function $f$ and $f^{\prime }\left(x\right)=\sin x$, then $g^{\prime }\left(x\right)=$

• A. $cosec\left\{g\right(x\left)\right\}$
• B. $\sin \left\{g\right(x\left)\right\}$
• C. $\frac{1}{\sin \left\{g\right(x\left)\right\}}$
• D. none of these

Answer 1

The correct option is C $\frac{1}{\sin \left\{g\right(x\left)\right\}}$

Since $g$ is the inverse of function $f$. Therefore,
$fog\left(x\right)=x\text{for all}x$
$\frac{d}{dx}\left(fog\right(x\left)\right)=\frac{d}{dx}\left(x\right)\text{for all}x$
$\frac{d}{dg\left(x\right)}\left\{fog\right(x\left)\right\}.g^{\prime }\left(x\right)=1\text{for all}x$
$\sin \left\{g\right(x\left)\right\}{g}^{\prime }\left(x\right)=1\text{for all}x$
$\therefore g^{\prime }\left(x\right)=\frac{1}{\sin \left\{g\right(x\left)\right\}}\text{for all}x$