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# product rule

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

$\frac{\mathrm{d}}{\mathrm{d}x}\left(f\cdot g\right)(x_{0})=f(x_{0})g^{{\prime}% }(x_{0})+f^{{\prime}}(x_{0})g(x_{0}).$ |

# Proof

See the proof of the product rule.

# 0.1 Generalized Product Rule

More generally, for differentiable functions $f_{1},f_{2},\ldots,f_{n}$ in one variable, all differentiable at $x_{0}$, we have

$\displaystyle D(f_{1}\cdots f_{n})(x_{0})=\sum_{{i=1}}^{n}\left(f_{i}(x_{0})% \cdots f_{{i-1}}(x_{0})\cdot Df_{i}(x_{0})\cdot f_{{i+1}}(x_{0})\cdots f_{n}(x% _{0})\right).$ |

Also see Leibniz’ rule.

# Example

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 $f^{{\prime}}(x)=1$ and $g^{{\prime}}(x)=\frac{1}{x}$. Therefore, by the product rule,

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}(x\ln|x|)$ | $\displaystyle=$ | $\displaystyle f(x)g^{{\prime}}(x)+f^{{\prime}}(x)g(x)$ | ||

$\displaystyle=$ | $\displaystyle\frac{x}{x}+1\cdot\ln|x|$ | |||

$\displaystyle=$ | $\displaystyle\ln|x|+1$ |

Related:

Derivative,ProofOfProductRule, ProductRule, PowerRule, ProofOfPowerRule, SumRule, ZeroesOfDerivativeOfComplexPolynomial

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

26A06*no label found*

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