alternative proof of derivative of $x^{n}$

The typical derivative formula

\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

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

for any positive integer $n$ .

Proof.

$\begin{array}[]{ll}\displaystyle\frac{d}{dx}(x^{n})&\displaystyle=\lim_{h\to 0% }\frac{(x+h)^{n}-x^{n}}{h}\\ &\\ &\displaystyle=\lim_{h\to 0}\frac{\displaystyle\left(\sum_{j=0}^{n}{n\choose j% }x^{j}h^{n-j}\right)-x^{n}}{h}\\ &\\ &\displaystyle=\lim_{h\to 0}\frac{\displaystyle x^{n}+nx^{n-1}h+h^{2}\left(% \sum_{j=0}^{n-2}{n\choose j}x^{j}h^{n-2-j}\right)-x^{n}}{h}\\ &\\ &\displaystyle=\lim_{h\to 0}\frac{\displaystyle nx^{n-1}h+h^{2}\sum_{j=0}^{n-2% }{n\choose j}x^{j}h^{n-2-j}}{h}\\ &\\ &\displaystyle=\lim_{h\to 0}\left(nx^{n-1}+h\sum_{j=0}^{n-2}{n\choose j}x^{j}h% ^{n-2-j}\right)\\ &\\ &\displaystyle=nx^{n-1}\end{array}$

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Title	alternative proof of derivative of $x^{n}$
Canonical name	AlternativeProofOfDerivativeOfXn
Date of creation	2013-03-22 15:59:31
Last modified on	2013-03-22 15:59:31
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	12
Author	Wkbj79 (1863)
Entry type	Proof
Classification	msc 26B05
Classification	msc 26A24
Related topic	DerivativeOfXn
Related topic	DerivativesByPureAlgebra

alternative proof of derivative of xn

Proof.

alternative proof of derivative of $x^{n}$