Product rule

Builds on Derivative Differentiable functions are continuous

Statement

Let $f$ and $g$ be real functions on an open interval $I$ , both differentiable at $a \in I$ . Then $f g$ is differentiable at $a$ , and $(f g)'(a) = f'(a) g(a) + f(a) g'(a).$

Remarks

Write $f(a + h) g(a + h) - f(a) g(a) = (f(a + h) - f(a)) g(a + h) + f(a) (g(a + h) - g(a))$ . Divide by $h$ and use the continuity of $g$ at $a$ . For example, $(x \sin x)' = \sin x + x \cos x$ . Mathlib states it as HasDerivAt.mul.

History

The discovery of the rule is credited to Gottfried Leibniz.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Limit point of a set definition
Limit of a function definition
Continuity at a point definition
Derivative definition
Differentiable function definition
Differentiable functions are continuous statement
Product rule statement

Tag	`0V9A43H7WV`
URI	`m2://0V9A43H7WV`
Aliases	`leibniz-rule-for-products`
Kind	statement, theorem
Writing	W1 stated
MSC	26A24
Status	This copy of the site has no build data. Run `m2 status 0V9A43H7WV`.
Uses	Derivative, Differentiable functions are continuous
Same as	Wikidata Q466720 Wikipedia Product rule

Product rule

Statement

Remarks

History

Reading path

Where this leads

Further reading

What this page needs for W2

What links here

Used by