PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
Owner confidence rating: Medium Entry average rating: No information on entry rating
product rule (Theorem)

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

$\displaystyle \frac{\ensuremath{\mathrm{d}}}{\ensuremath{\mathrm{d}x}}\left(f\cdot g\right)(x_0) = f(x_0)g'(x_0) + f'(x_0)g(x_0).$    

Proof

See the proof of the product rule.

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\vert x\vert$ can be found by application of this rule. Let $ f(x) = x, g(x) = \ln\vert x\vert$, so that $ f(x)g(x) = x\ln\vert x\vert$. Then $ f'(x) = 1$ and $ g'(x) = \frac{1}{x}$. Therefore, by the product rule,


$\displaystyle \frac{\ensuremath{\mathrm{d}}}{\ensuremath{\mathrm{d}x}}(x\ln\vert x\vert)$ $\displaystyle =$ $\displaystyle f(x)g'(x) + f'(x)g(x)$  
  $\displaystyle =$ $\displaystyle \frac{x}{x} + 1\cdot\ln\vert x\vert$  
  $\displaystyle =$ $\displaystyle \ln\vert x\vert + 1$  



"product rule" is owned by mathcam. [ full author list (2) | owner history (1) ]
(view preamble)

View style:

See Also: derivative, proof of product rule, product rule, power rule, proof of the power rule, sum rule


Attachments:
proof of product rule (Proof) by mathcam
logarithmic proof of product rule (Proof) by Wkbj79
Log in to rate this entry.
(view current ratings)

Cross-references: differentiable functions, product, derivative, point, differentiable, variable, functions
There are 26 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 8126 times total.

Classification:
AMS MSC26A06 (Real functions :: Functions of one variable :: One-variable calculus)

Pending Errata and Addenda
None.
[ View all 4 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | prove | add result | add corollary | add example | add (any)