Question

Let $f\left(x\right)=e^{x+sinx}$ and $g\left(x\right)=f^{-1}\left(x\right),h\left(x\right)=g\left(x\right)+g^{\prime }\left(x\right)$ then $4h\left(1\right)$ is : ___

Answer 1

$f^{\prime }\left(x\right)=e^{x+sinx}(1+cosx)$
$f\left(0\right)=1\Rightarrow f^{-1}\left(1\right)=0\Rightarrow g\left(1\right)=0$
$\therefore h\left(1\right)=g\left(1\right)+g^{\prime }\left(1\right)=0+\frac{1}{f^{\prime }\left(0\right)}=\frac{1}{2}$
$\therefore 4h\left(1\right)=2$