Let \(f\) be an invertible function, let \(x\) be in the domain of \(f\), and let \(y\) be in the codomain of \(f\). Since \(f\) is an invertible function, we know that:
\[ f(f^{-1}(y))=y\]
Differentiate both sides:
\[ \frac{dy}{dy} = f'(f^{-1}(y)) * (f^{-1})'(y) \]
This means:
\[ 1 = f'(f^{-1}(y)) * (f^{-1})'(y) \]
Rearranging:
\[ (f^{-1})'(y) = \frac{1}{f'(f^{-1}(y))} \]