Inverse Rule (Analysis)

analysis

Definition

Inverse Rule

Let $I,J\subseteq \mathbb{R}$ be open intervals and let $f:I\to J$ be a bijective differentiable function with $f^{\prime }(x)\ne 0$ for all $x\in I$. Then the inverse function $f^{-1}:J\to I$ is differentiable and its derivative at $y\in J$ is:

$$(f^{-1}{)}^{\prime }(y)=\frac{1}{f^{\prime }(f^{-1}(y))}$$

In Leibniz notation, with $y=f(x)$ and $x=f^{-1}(y)$:

$$\frac{dx}{dy}=\frac{1}{\frac{dy}{dx}}$$

Intuition

Reciprocal slopes

The derivative of $f$ measures how much $y$ changes per unit change in $x$. The inverse $f^{-1}$ swaps the roles of $x$ and $y$, so its rate of change is the reciprocal. If $f$ rises steeply (large $f^{\prime }$), then $f^{-1}$ crawls horizontally (small $(f^{-1}{)}^{\prime }$), and vice versa.

Geometrically, the graph of $f^{-1}$ is the reflection of the graph of $f$ across the line $y=x$. Reflecting a tangent segment swaps its horizontal and vertical components, turning a slope of $\frac{\Delta y}{\Delta x}$ into $\frac{\Delta x}{\Delta y}$.