# 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 (http://planetmath.org/ProofOfProductRule).

## 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 (http://planetmath.org/LeibnizRule).

### 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,

 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}(x\ln|x|)$ $\displaystyle=$ $\displaystyle f(x)g^{\prime}(x)+f^{\prime}(x)g(x)$ $\displaystyle=$ $\displaystyle\frac{x}{x}+1\cdot\ln|x|$ $\displaystyle=$ $\displaystyle\ln|x|+1$
 Title product rule Canonical name ProductRule Date of creation 2013-03-22 12:27:57 Last modified on 2013-03-22 12:27:57 Owner mathcam (2727) Last modified by mathcam (2727) Numerical id 12 Author mathcam (2727) Entry type Theorem Classification msc 26A06 Related topic Derivative Related topic ProofOfProductRule Related topic ProductRule Related topic PowerRule Related topic ProofOfPowerRule Related topic SumRule Related topic ZeroesOfDerivativeOfComplexPolynomial