Let $g\left(x\right)$ be the inverse of an inversible function $f\left(x\right)$ which is differentiable at $x=c$ then $g^1\left(f\left(c\right)\right)=$

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

$g\left(f\left(c\right)\right)=f^{-1}\left(f\left(c\right)\right)$

$=c$

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

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

Differentiating with receptive to x

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

$x=f\left(c\right)$

$f^1(g\left(f\left(c\right)\right).g^1\left(f\left(c\right)\right)=1$

$f^1\left(c\right).g^1\left(f\left(c\right)\right)=1$

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