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The product rule states that if $f:\mathbb{R}\rightarrow\mathbb{R}$ and $g:\mathbb{R}\rightarrow\mathbb{R}$ are functions in one variable both differentiable at a point $x_0$ , then the derivative of the product of the two functions, denoted $f\cdot g$ , at $x_0$ is given by
See the proof of the product rule.
More generally, for differentiable functions $f_1, f_2,\ldots,f_n$ in one variable, all differentiable at $x_0$ , we have
Also see Leibniz' rule.
The derivative of $x\ln|x|$ can be found by application of this rule. Let $f(x) = x, g(x) = \ln|x|$ , so that $f(x)g(x) = x\ln|x|$ . Then $f'(x) = 1$ and $g'(x) = \frac{1}{x}$ . Therefore, by the product rule,
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Cross-references: application, differentiable functions, product, derivative, point, differentiable, variable, functions
There are 30 references to this entry.
This is version 9 of product rule, born on 2002-02-24, modified 2007-01-11.
Object id is 2628, canonical name is ProductRule.
Accessed 10725 times total.
Classification:
| AMS MSC: | 26A06 (Real functions :: Functions of one variable :: One-variable calculus) |
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Pending Errata and Addenda
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