If g is the inverse of function f and f^ (x)= x , then ^ (x) is...

Since, $g$ is the inverse of function $f$. Therefore, $\mathrm{g}(\mathrm{x})=\mathrmf^{-1}(\mathrm{x})\Rightarrow \mathrm{f}[\mathrm{g}(\mathrm{x})]=\mathrm{x}\Rightarrow$ fog $(\mathrm{x})=\mathrm{x},$ for all $\mathrm{x}$ Differentiate both side,w.r.tx $\Rightarrow \frac{\mathrm{d}}{\mathrm{dx}}\{\operatorname{fog}(\mathrm{x})\}=\frac{\mathrm{d}}{\mathrm{dx}}(\mathrm{x}),$ for all $\mathrm{x}\Rightarrow \mathrmf^{\prime}[\mathrm{g}(\mathrm{x})] \mathrmg^{\prime}(\mathrm{x})=1,$ for all $\mathrm{x}\Rightarrow \sin \{\mathrm{g}(\mathrm{x})\} \mathrmg^{\prime}(\mathrm{x})=1,$ for all $\mathrm{x}$ (By defn of \left.f^{\prime}(x)\right)\Rightarrow \mathrmg^{\prime}(\mathrm{x})=\frac{1}{\sin \{\mathrm{g}(\mathrm{x})\}}