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power rule

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

 Equations $$\frac{d}{dx} x^{n} = nx^{n-1}$$

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 Breakdown Mathematics > Calculus/Analysis >> Calculus

 Extended explanation Power rule applies to a function of the form $x^{n}$, 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: $$f'(x) := \lim_{h→0} \frac{f(x+h)-f(x)}{h}$$ Substituting in gives us: $$\frac{d}{dx} x^{n} = \lim_{h→0} \frac{(x+h)^{n}-x^{n}}{h}$$ If we then expand using Binomial Theorem: $$\frac{d}{dx} x^{n} = \lim_{h→0} \frac{x^{n}+nx^{n-1}h+\binom{n}{2}x^{n-2}h^{2}+\cdots+h^{n} -x^{n}}{h}$$ We can then cancel the first and last $x^{n}$ terms and distribute the h from the denominator: $$\frac{d}{dx} x^{n} = \lim_{h→0} nx^{n-1}+\binom{n}{2}x^{n-2}h+\cdots+h^{n-1}$$ Finally, we take the limit by substituting in h=0: $$\frac{d}{dx} x^{n} = nx^{n-1}+\binom{n}{2}x^{n-2}0+\cdots+0^{n-1}$$ $$\frac{d}{dx} x^{n} = nx^{n-1}$$ Example 1: $$f(x) = x^{189}$$ $$f'(x) = 189x^{189-1} = 189x^{188}$$ Example 2: $$f(x) = 3x^{3}+7x^{2}+8x+2$$ $$f'(x) = 9x^{2}+14x+8$$ Example 3: $$f(x) = 3\sqrt{x}$$ $$f'(x) = 3×1/2\ x^{(1/2-1)} = \frac{3}{2\sqrt{x}}$$