Power rule

Polynomials are perhaps the simplest functions to do calculus with. There are only 2 basic things you need to know: how to differentiate polynomials, and how to integrate them.

These are given by the following two simple rules:

${\frac {d}{dx}}\sum _{r=m}^{n}a_{r}x^{r}=\sum _{r=m}^{n}a_{r}\times r\times x^{r-1}$

$\int \sum _{r=m}^{n}a_{r}x^{r}=\sum _{r=m}^{n}{\frac {a^{r}}{r+1}}x^{r+1}$

Because differentiation is linear, we have:

${\frac {d\left(\sum _{r=0}^{n}a_{r}x^{r}\right)}{dx}}=\sum _{r=0}^{n}{\frac {d\left(a_{r}x^{r}\right)}{dx}}=\sum _{r=0}^{n}a_{r}{\frac {d\left(x^{r}\right)}{dx}}$

So it remains to find ${\frac {d\left(x^{r}\right)}{dx}}$ for any natural number r. By the binomial theorem,

$\left(x+h\right)^{r}=\sum _{k=0}^{r}C\left(r,k\right)x^{k}h^{r-k}$

and therefore

${\frac {\left(x+h\right)^{r}-x^{r}}{h}}=\sum _{k=0}^{r-1}C\left(r,k\right)x^{k}h^{r-k-1}$

All but one term in this sum contain an h, and therefore the limit as h→0 is equal to the term with k=r-1:

$C\left(r,r-1\right)x^{r-1}=rx^{r-1}$

This limit is equal to the derivative of x^r by definition.