# 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.

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

Title alternative proof of derivative of $x^{n}$ AlternativeProofOfDerivativeOfXn 2013-03-22 15:59:31 2013-03-22 15:59:31 Wkbj79 (1863) Wkbj79 (1863) 12 Wkbj79 (1863) Proof msc 26B05 msc 26A24 DerivativeOfXn DerivativesByPureAlgebra