If the function $f(x)=x^3+e^{\frac{x}{2}}$ and $g(x)=f^{-1}(x)$, then the value of $g^{\prime}(1)$ is

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Aicharya · Accepted Answer

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

If the function $f(x)=x^3+e^{\frac{x}{2}}$ and $g(x)=f^{-1}(x)$, then the value of $g^{\prime}(1)$ is

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