Question

If $g$ is inverse of $f\left(x\right)=x^3+x+\cos x$, then find the value of $g^{\prime }\left(1\right)$.

Answer 1

If $g$ is inverse of $f\left(x\right)$ the

$g\left(f\right(x\left)\right)=1$

$g^1\left(f\right(x\left)\right)=\frac{1}{f^1\left(x\right)}$

when $x=0f\left(x\right)=1$

$g^1\left(1\right)=\frac{1}{f^1\left(0\right)}\Rightarrow f^1\left(10\right)={30}^2+1+\sin \theta$

$=1$ $=1$