inverse function rule
Inverse function rule
If \[f(x)\] is both invertible and differentiable, it seems reasonable that the inverse of \[f(x)\], written as \[f^{-1}(x)\] is also differentiable. The inverse function rule is written as \[(f^{-1})^{\prime}(x)=\frac{1}{f^{\prime}(f^{-1}(x))}\].
Proof
Since from the graph we see that \[f^{\prime}(f^{-1}(x))=\frac{q}{p}\] and \[(f^{-1})^{\prime}(x)=\frac{p}{q}\], thus \[(f^{-1})^{\prime}(x)=\frac{1}{f^{\prime}(f^{-1}(x))}\], proving the formula.