Mathematics - Differentiation Question with Solution | TestHub

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