PlanetMath: alternative proof of derivative of $x^n$

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alternative proof of derivative of $x^n$

(Proof)

The typical derivative formula

$\displaystyle \frac{df}{dx}=\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}$

combined with the binomial theorem yield an alternative way to prove that

$\displaystyle \frac{d}{dx}(x^n)=nx^{n-1}$

for any positive integer .

Proof.

$\begin{displaymath}\begin{array}{ll} \displaystyle \frac{d}{dx}(x^n) & \displays... ...n-2-j} \right) \ & \ & \displaystyle = nx^{n-1} \end{array}\end{displaymath}$

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See Also: derivative of $x^n$

, derivatives by pure algebra

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Cross-references: integer, positive, binomial theorem, derivative
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This is version 9 of alternative proof of derivative of $x^n$

, born on 2006-06-11, modified 2007-06-07.
Object id is 8013, canonical name is AlternativeProofOfDerivativeOfXn.
Accessed 2032 times total.

Classification:

AMS MSC:	26A24 (Real functions :: Functions of one variable :: Differentiation : general theory, generalized derivatives, mean-value theorems)
	26B05 (Real functions :: Functions of several variables :: Continuity and differentiation questions)

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