Question

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

Answer 1

Given $g\left(x\right)=f^{-1}\left(x\right)\Rightarrow g\left(f\right(x\left)\right)=x$
Differentiating w.r.t $x$
$g^{\prime }\left(f\right(x\left)\right).f^{\prime }\left(x\right)=1$
$\Rightarrow g^{\prime }\left(f\right(x\left)\right)=\frac{1}{f^{\prime }\left(x\right)}$
Here $f\left(0\right)=1\Rightarrow g^{\prime }\left(1\right)=\frac{1}{f^{\prime }\left(0\right)}=1$