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What is the product rule

  1. Jul 23, 2014 #1
    Definition/Summary

    The product rule is a method for finding the derivative of a product of functions.

    Equations

    [tex](fg)'\ =\ f'g\ +\ fg'[/tex]

    [tex](fgh)'\ =\ f'gh\ +\ fg'h\ +\ fgh'[/tex]

    Extended explanation

    If a function F is the product of two other functions f and g (i.e. F(x) = f(x)g(x)), then the product rules states that:

    [tex]\frac{d}{dx}F \ = \ \frac{df}{dx}g \ + \ f\frac{dg}{dx}[/tex]

    Proof:

    [tex]\frac{F(x + h) - F(x)}{h} \ = \ \frac{f(x + h)g(x + h) - f(x)g(x)}{h}[/tex]

    [tex] = \ \frac{f(x + h)g(x + h) \ - \ f(x)g(x + h) \ + \ f(x)g(x + h) \ - \ f(x)g(x)}{h}[/tex]

    [tex] = \ \frac{f(x + h) - f(x)}{h}g(x + h)
    \ + \ f(x)\frac{g(x+h) - g(x)}{h}[/tex]

    Now take the limit as h approaches zero.

    [tex]\frac{d}{dx}F \ = \ \lim_{h \to 0}\frac{F(x + h) - F(x)}{h}
    \ = \ \lim_{h \to 0}\frac{f(x + h) - f(x)}{h}g(x + h)
    \ + \ \lim_{h \to 0}f(x)\frac{g(x+h) - g(x)}{h}
    \ = \ \frac{df}{dx}g \ + \ f\frac{dg}{dx}[/tex]

    * This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
     
  2. jcsd
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