Quotient Rule (Analysis)

analysis

Definition

Quotient Rule

Let $I\subseteq \mathbb{R}$ be an open interval and let $f,g:I\to \mathbb{R}$ be differentiable functions with $g(x)\ne 0$ for all $x\in I$. Then the quotient $h(x)=\frac{f(x)}{g(x)}$ is differentiable on $I$ and its derivative is:

$$h^{\prime }(x)=\frac{f^{\prime }(x)\cdot g(x)-f(x)\cdot g^{\prime }(x)}{[g(x){]}^2}$$

In Leibniz notation:

$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{\frac{df}{dx}\cdot g(x)-f(x)\cdot \frac{dg}{dx}}{[g(x){]}^2}$$

Intuition

Competing effects

Regard $\frac{f}{g}$ as a ratio. A small change $\Delta x$ changes the numerator by $\Delta f\approx f^{\prime }(x)\Delta x$ and the denominator by $\Delta g\approx g^{\prime }(x)\Delta x$. The change in the ratio has two opposing contributions:

• A growing numerator pulls the ratio up (scaled by the denominator).
• A growing denominator pushes the ratio down (scaled by the numerator).

The net effect is their difference, normalised by the squared denominator to account for the scale:

$$\Delta \left(\frac{f}{g}\right)\approx \frac{f^{\prime }(x)\Delta x\cdot g(x)-f(x)\cdot g^{\prime }(x)\Delta x}{[g(x){]}^2}$$

Dividing by $\Delta x$ and taking the limit gives the quotient rule.