Differentiation rules > Chain rule

A composed function is a function consisting of two or more functions that are connected in series.

The derivative of a composed function can be found with

Differentiation rule 4 (chain rule):

If
$S\left(x\right)=f\left(g\right(x\left)\right)$ then
$S\text{'}\left(x\right)=f\text{'}\left(g\right(x\left)\right)\cdot g\text{'}\left(x\right)$.

You can use this rule to prove the power rule for root functions:

If $f\left(x\right)=\sqrt[n]{x}={x}^{\frac{1}{n}}$ then $f\text{'}\left(x\right)=\frac{1}{n}{x}^{\frac{1}{n}-1}$ for positive integer $n$ and $x\ne 0$.

In general:

Differentiation rule 5 (general power rule):

If
$f\left(x\right)={x}^{r}$ then
$f\text{'}\left(x\right)=r{x}^{r-1}$ for any real value of
$r$.

To be able to prove the differentiation rule, you first need to learn how to differentiate exponential functions and logaritmic functions. That will come later.