# binomial proof of positive integer power rule

We will use the difference quotient in this proof of the power rule for positive integers. Let $f(x)=x^{n}$ for some integer $n\geq 0$. Then we have

 $\displaystyle f^{\prime}(x)=\lim_{h\rightarrow 0}\frac{(x+h)^{n}-x^{n}}{h}.$

We can use the binomial theorem to expand the numerator

 $\displaystyle f^{\prime}(x)=\lim_{h\rightarrow 0}\frac{C_{0}^{n}x^{0}h^{n}+C_{% 1}^{n}x^{1}h^{n-1}+\cdots+C_{n-1}^{n}x^{n-1}h^{1}+C_{n}^{n}x^{n}h^{0}-x^{n}}{h}$

where $C_{k}^{n}=\frac{n!}{k!(n-k)!}$. We can now simplify the above

 $\displaystyle f^{\prime}(x)$ $\displaystyle=\lim_{h\rightarrow 0}\frac{h^{n}+nxh^{n-1}+\cdots+nx^{n-1}h+x^{n% }-x^{n}}{h}$ $\displaystyle=\lim_{h\rightarrow 0}(h^{n-1}+nxh^{n-2}+\cdots+nx^{n-1})$ $\displaystyle=nx^{n-1}$ $\displaystyle=nx^{n-1}.$
Title binomial proof of positive integer power rule BinomialProofOfPositiveIntegerPowerRule 2013-03-22 12:29:43 2013-03-22 12:29:43 mathcam (2727) mathcam (2727) 8 mathcam (2727) Proof msc 26A03