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3. Power Rule

The power rule states that for any number \(a,\) \[\frac{d}{dx}x^a = ax^{a-1}\]

Proof for positive integers \(n\):
Let \(n\) be a positive integer. By definition of derivatives, \[\frac{d}{dx}x^n = \lim_{h \rightarrow 0} \frac{(x+h)^n - x^n}{h}\] The Binomial Theorem states that for any positive integer \(n,\) \[(x+h)^n = \sum_{i=0}^n {n \choose i}x^i h^{n-i}\] If we separate the terms in the series when \(i=n\) and \(i=n-1,\) we get \begin{align} (x+h)^n & = {n \choose n}x^n + {n \choose n-1}x^{n-1}h + \sum_{i=0}^{n-2} {n \choose i}x^i h^{n-i} \\ & = x^n + nx^{n-1}h + \sum_{i=0}^{n-2} {n \choose i}x^i h^{n-i} \\ \end{align} Plugging in this representation for \((x+h)^n,\) we get \begin{align} \frac{d}{dx}x^n & = \lim_{h \rightarrow 0} \frac{x^n + nx^{n-1}h + \sum_{i=0}^{n-2} {n \choose i}x^i h^{n-i} - x^n}{h} \\ & = \lim_{h \rightarrow 0} \frac{nx^{n-1}h + \sum_{i=0}^{n-2} {n \choose i}x^i h^{n-i}}{h} \\ & = \lim_{h \rightarrow 0} \left[nx^{n-1} + \sum_{i=0}^{n-2} {n \choose i}x^i h^{n-i-1}\right] \\ & = nx^{n-1} \end{align}

Examples

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