1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

What is the power rule

  1. Jul 23, 2014 #1
    Definition/Summary

    A method used to take the derivative of a polynomial function.

    Equations

    [tex]\frac{d}{dx} x^{n} = nx^{n-1}[/tex]

    Extended explanation

    Power rule applies to a function of the form [itex]x^{n}[/itex], where x is the variable and n is a constant. Used in combination with the sum and constant factor rules of differentiation, power rule can be a powerful tool for taking derivatives.

    Proof:

    We can apply the limit definition of a derivative to this specific function:
    [tex]f'(x) := \lim_{h→0} \frac{f(x+h)-f(x)}{h}[/tex]
    Substituting in gives us:
    [tex]\frac{d}{dx} x^{n} = \lim_{h→0} \frac{(x+h)^{n}-x^{n}}{h}[/tex]
    If we then expand using Binomial Theorem:
    [tex]\frac{d}{dx} x^{n} = \lim_{h→0} \frac{x^{n}+nx^{n-1}h+\binom{n}{2}x^{n-2}h^{2}+\cdots+h^{n} -x^{n}}{h}[/tex]
    We can then cancel the first and last [itex]x^{n}[/itex] terms and distribute the h from the denominator:
    [tex]\frac{d}{dx} x^{n} = \lim_{h→0} nx^{n-1}+\binom{n}{2}x^{n-2}h+\cdots+h^{n-1}[/tex]
    Finally, we take the limit by substituting in h=0:
    [tex]\frac{d}{dx} x^{n} = nx^{n-1}+\binom{n}{2}x^{n-2}0+\cdots+0^{n-1}[/tex]
    [tex]\frac{d}{dx} x^{n} = nx^{n-1}[/tex]

    Example 1:
    [tex]f(x) = x^{189}[/tex]
    [tex]f'(x) = 189x^{189-1} = 189x^{188}[/tex]
    Example 2:
    [tex]f(x) = 3x^{3}+7x^{2}+8x+2[/tex]
    [tex]f'(x) = 9x^{2}+14x+8[/tex]
    Example 3:
    [tex]f(x) = 3\sqrt{x}[/tex]
    [tex]f'(x) = 3×1/2\ x^{(1/2-1)} = \frac{3}{2\sqrt{x}}[/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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted