Product rule in several variables

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 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, 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 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 $(x, y)$.