Geometrically understanding calculus of inverse functions (2023)

Summary

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 \(f'(a) \neq 0\) at some point, the inverse function theorem states that there is some interval \(I\) with \(a \in I\) such that there exists a continuously differentiable inverse \(f^{-1}\) defined on \(f(I)\) such that for all \(x \in I\) \[\frac{df^{-1}}{dx}(x) = \frac{1}{f'(f^{-1}(x))}.\] A simple example of the theorem in action is finding the derivative of \(\ln x\), which evaluates to \(1/e^{\ln x} = 1/x\). The standard high school approach to deduce the above in high school is to differentiate both sides of \(f^{-1}(f(x)) = x\). This formal approach is quite dry and things get a lot more interesting when we think geometrically. Geometrically, given some function \(f(x) = y\) the inverse is just taking the plot of the function and reflecting it about the diagonal line \(y=x\). As such all tangent lines are also reflected along the diagonal line and hence the slope is inversed. Below we have the graph of \(e^x\) (the blue line) and the graph of \(\ln x\) (the red line). And you can see how the tangent line of \(e^x\) at \((2,e^2)\) is reflected along \(y=x\) to give the tangent line of \(\ln x\) at \((...