Product rule in several variables

Builds on Differentiable function of several variables Fréchet derivative

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.

Reading path

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Euclidean space object
Function concept
Fréchet derivative definition
Function of several variables concept
Limit of a function of several variables concept
Differentiable function of several variables definition
Product rule in several variables statement

Tag	`0SF85PY0QN`
URI	`m2://0SF85PY0QN`
Kind	statement, proposition
Writing	W0 stub
MSC	26B05
Status	This copy of the site has no build data. Run `m2 status 0SF85PY0QN`.
Uses	Differentiable function of several variables, Fréchet derivative

Product rule in several variables

Statement

Remarks

Reading path

Further reading

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