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Power rule for derivatives

For a power function $f(x)=ax^n$ with $a,n\in \mathbb{R}$ and the exponent $n\ne 0$, the derivative is given by:

[math]f'(x)=anx^{n-1}[/math]

Prerequisite information[edit]

The proof[edit]

From the definition of the derivative:

[math]\begin{eqnarray} \frac{d}{dx}ax^n &=& \lim_{h\to 0} a\frac{(x+h)^n-x^n}{h}\end{eqnarray}[/math]

We use the Binomial theorem,

$\begin{eqnarray}(x+h)^n &=& x^n + n x^{n-1}h + O(h^2)\end{eqnarray}$, where $O(h^2)$ has terms in $h^2$ and higher.

Then we have

[math]\begin{eqnarray} \frac{d}{dx}ax^n &=& a\frac{d}{dx}x^n \\ &=& a\lim_{h\to 0} \frac{x^n + n x^{n-1}h + O(h^2) - x^n}{h} \\ &=& a\lim_{h\to 0} \frac{n x^{n-1}h + O(h^2)}{h} \\ &=& a\lim_{h\to 0} (n x^{n-1} + O(h)) \\ &=& an x^{n-1}\end{eqnarray}[/math]