Question

g(x) = x^5 + 4x^3 + x + 1 , find (g^-1) ' (7)

Answer 1

Let
\[f =g^{-1}\\
f(g(x)) =x\\
f'(g(x)) g'(x) =1\\
Notice \, that \\
g(1)=7\\
f'(g(1)) g'(1) =1\\
f'(7) = \frac 1 {g'(1)}
\]
Compute the last line and you are done.