Let $ < < < $. If $ f: ( , ) $ is continously differentiable and $ f ’ (x_0) $ for some $ x_0 (, )$, then f is invertible in a neighbourhood of $ x_0 $ and $ f ^{-1} $ is a differentiable function such that $$ (f^{-1})'(y_0) = \frac{1}{f'(x_0)} $$ where $ y_0 = f(x_0)$
If f is continous and injective in a neighbourhood of x0 then this formula also works
$(f^-1)'(y_0) = \frac{1}{f'(f^-1(y_0))}$ ## intuition If f is continous and injective in a neighbourhood of x0 then this formula also works ## rigour Let $ < < < $. If $ f: ( , ) $ is continously differentiable and $ f ’ (x_0) $ for some $ x_0 (, )$, then f is invertible in a neighbourhood of $ x_0 $ and $ f ^{-1} $ is a differentiable function such that $$ (f^{-1})'(y_0) = \frac{1}{f'(x_0)} $$ where $ y_0 = f(x_0)$ %% ==exam clinic %% ==examples and non-examples %% ==resources