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power rule
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Definition/Summary
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| A method used to take the derivative of a polynomial function. |
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Equations
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| [tex]\frac{d}{dx} x^{n} = nx^{n-1}[/tex] |
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Recent forum threads on power rule
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Breakdown
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Mathematics
> Calculus/Analysis
>> Calculus
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Extended explanation
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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] |
Commentary
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