Mathematics - Differentiation Question with Solution | TestHub

Given, $g\{f(x)\}=x$ $$\begin{array}{lc}\Rightarrow & g^{\prime }\{f(x)\}{f}^{\prime }(x)=1\\ \text{ If }& f(x)=1\Rightarrow x=0,f(0)=1\end{array}$$ Substitute $x=0$ in Eq. (i), we get $$\begin{array}{rl}& g^{\prime }(1)=\frac{1}{f^{\prime }(0)}\\ & \Rightarrow g^{\prime }(1)=2\\ & \left[\because f^{\prime }(x)=3x^2+\frac{1}{2}{e}^{x/2}\Rightarrow f^{\prime }(0)=\frac{1}{2}\right]\\ & \end{array}$$