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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

• The power rule can be used to compute the derivative of $$x^2$$ as follows: \begin{align} \frac{d}{dx}x^2 & = 2x^{2-1} \\ & = 2x \end{align}
• Writing $$x$$ as $$x^1,$$ the power rule can be used to compute the derivative of $$x$$ as follows: \begin{align} x & = x^1 \\ & = 1x^{1-1} \\ & = 1 \end{align}
• Writing $$\frac{1}{x^3}$$ as $$x^{-3},$$ the power rule can be used to compute the derivative of $$\frac{1}{x^3}$$ as follows: \begin{align} \frac{1}{x^3} & = x^{-3} \\ & = -3x^{-3-1} \\ & = -3x^{-4} \\ & = -\frac{3}{x^4} \end{align}
• Writing $$\sqrt{x}$$ as $$x^{1/2},$$ the power rule can be used to compute the derivative of $$\sqrt{x}$$ as follows: \begin{align} \sqrt{x} & = x^{1/2} \\ & = \frac{1}{2}x^{1/2-1} \\ & = \frac{1}{2}x^{-1/2} \\ & = \frac{1}{2\sqrt{x}} \end{align}