Product Rule (Analysis)

analysis

Definition

Product Rule

Let $I\subseteq \mathbb{R}$ be an open interval and let $f,g:I\to \mathbb{R}$ be differentiable functions. Then the product $h(x)=f(x)\cdot g(x)$ is differentiable on $I$ and its derivative is:

$$h^{\prime }(x)=f^{\prime }(x)\cdot g(x)+f(x)\cdot g^{\prime }(x)$$

In Leibniz notation:

$$\frac{d}{dx}[f(x)\cdot g(x)]=\frac{df}{dx}\cdot g(x)+f(x)\cdot \frac{dg}{dx}$$

Intuition

Area of a rectangle

Regard $f(x)$ and $g(x)$ as the side lengths of a rectangle. A small change $\Delta x$ produces changes $\Delta f\approx f^{\prime }(x)\Delta x$ and $\Delta g\approx g^{\prime }(x)\Delta x$. The change in area is the sum of two strips, neglecting the vanishingly small corner:

$$\Delta (fg)\approx f^{\prime }(x)\Delta x\cdot g(x)+f(x)\cdot g^{\prime }(x)\Delta x$$

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