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26A03-BinomialProofOfPositiveIntegerPowerRule.tex
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26A03-BinomialProofOfPositiveIntegerPowerRule.tex
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\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{BinomialProofOfPositiveIntegerPowerRule}
\pmcreated{2013-03-22 12:29:43}
\pmmodified{2013-03-22 12:29:43}
\pmowner{mathcam}{2727}
\pmmodifier{mathcam}{2727}
\pmtitle{binomial proof of positive integer power rule}
\pmrecord{8}{32721}
\pmprivacy{1}
\pmauthor{mathcam}{2727}
\pmtype{Proof}
\pmcomment{trigger rebuild}
\pmclassification{msc}{26A03}
\endmetadata
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\begin{document}
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
\begin{align*}
f'(x) = \lim_{h\rightarrow 0} \frac{(x+h)^n - x^n}{h}.
\end{align*}
We can use the binomial theorem to expand the numerator
\begin{align*}
f'(x) = \lim_{h\rightarrow 0} \frac{C_0^n x^0h^n + C_1^n x^1h^{n-1} + \cdots + C_{n-1}^n x^{n-1}h^1 + C_n^n x^nh^0 - x^n}{h}
\end{align*}
where $C_k^n=\frac{n!}{k!(n-k)!}$. We can now simplify the above
\begin{align*}
f'(x)&= \lim_{h\rightarrow 0} \frac{h^n + nxh^{n-1} + \cdots + nx^{n-1}h + x^n - x^n}{h}\\
&=\lim_{h\rightarrow 0} (h^{n-1} + nxh^{n-2} + \cdots + nx^{n-1})\\
&=nx^{n-1}\\
&= nx^{n-1}.
\end{align*}
%%%%%
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\end{document}