---
tag: 0SF85PY0QN
kind: statement
role: proposition
name: Product rule in several variables
msc: [26B05]
uses: [0135XZZQ8M, 08EVNYZF42]
sources:
  - url: https://en.wikipedia.org/wiki/Product_rule
    locator: Banach space
  - url: https://en.wikipedia.org/wiki/Product_rule
    locator: In vector calculus
---

## Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f, g \colon U \to \mathbb{R}$, and let $a \in U$.
Suppose that $f$ and $g$ are [[differentiable|0135XZZQ8M]] at $a$.
Then the product $f g$ is differentiable at $a$, and its derivative is
$$D(f g)(a) = f(a) \, Dg(a) + g(a) \, Df(a).$$
That is, $D(f g)(a)(h) = f(a) \, Dg(a)(h) + g(a) \, Df(a)(h)$ for each $h \in \mathbb{R}^n$.

## Remarks

In terms of the [[gradient|0ABK9V9Y64]], the rule reads $\nabla (f g) = f \nabla g + g \nabla f$.
The same rule holds for each partial derivative: $\partial_i (f g) = f \, \partial_i g + g \, \partial_i f$.
More generally, let $B$ be a bounded bilinear map. If $f$ and $g$ are [[Fréchet differentiable|08EVNYZF42]] at $a$, then $D(B(f, g))(a)(h) = B(Df(a) h, g(a)) + B(f(a), Dg(a) h)$. This gives product rules for the dot product and the cross product.

## Explanations

### L1 human

This is the familiar rule $(f g)' = f' g + f g'$, applied to the change in each direction at once.
For $f = x$ and $g = y$ on $\mathbb{R}^2$, the product $x y$ has the derivative $h \mapsto y h_1 + x h_2$ at the point $(x, y)$.