The Setting Recalled: Invertibility and Geometric Motivation
Recall: When Does a Function Have an Inverse?
From our earlier study of inverse functions, a function $f$ is invertible (one-to-one) on an interval $I$ precisely when it is strictly monotonic on $I$ — either strictly increasing or strictly decreasing throughout. If $f$ is also continuous on $I$, then the Continuous Inverse Theorem guarantees that $f^{-1}$ is also continuous on $f(I)$.
These two properties — strict monotonicity and continuity — are the prerequisites that make the derivative formula for the inverse possible.
The Reflection Principle
The graph of $f^{-1}$ is the reflection of the graph of $f$ across the line $y = x$. When you reflect a curve, you also reflect its tangent lines. A tangent line with slope $m \neq 0$ reflects to a tangent line with slope $\frac{1}{m}$.
The blue curve $f(x)=x^2$ and its inverse $f^{-1}(x)=\sqrt{x}$ (red) are reflections across $y=x$. Their tangent slopes at corresponding points are reciprocals.
The Inverse Function Theorem (Derivative Form)
Let $I$ be an open interval containing $x_0$, and let $f: I \to \mathbb{R}$ be strictly monotonic and continuous. If $f$ is differentiable at $x_0$ and $f'(x_0) \neq 0$, then the inverse function $f^{-1}$ is differentiable at $y_0 = f(x_0)$, and: