Geometrically understanding calculus of inverse functions (2023)

Given a function such as \(\tan x\), could you write \(\frac{d}{dx} \arctan x\) and \(\int \arctan x \; dx\), just from \(\tan x\), \(\frac{d}{dx} \tan x\) and \(\int \tan x \; dx\)? With some caveats, the inverse function theorem answers the former while the Legendre transformation answers the later. We’ll approach this with as much geometric intuition as possible, avoiding the dry application of formulas. Derivatives of inverse functions and the inverse function theorem Instead of approaching the inverse function theorem through formulas, we’ll explore it geometrically—it’s much more intuitive and enjoyable! But first, to refresh our memory, let’s revisit the formal statement of the inverse function theorem, which relates the derivative of \(f(x)\) and its inverse \(f^{-1}(x)\). Given a continuously differentiable function \(f: \mathbb{R} \to \mathbb{R}\) with \(f'(a) eq 0\) at some point, the inverse function theorem states that there is some interval \(I\) with \(a \in I\) suc ... Read full article.