The Chain Rule can also be applied to compound functions involving more than one function.
Given a function
$f\left(x\right)=\left(m\phantom{\rule{8px}{0ex}}\circ \phantom{\rule{8px}{0ex}}n\phantom{\rule{8px}{0ex}}\circ \phantom{\rule{8px}{0ex}}o\right)x$
The approach for this sequence of three functions is the same as for two functions. The key is to correctly identify the 'layers' of compound functions. Once the individual functions are identified, the result is the product of terms that take the derivatives of the outer functions and multiply each by the 'undisturbed' values of the functions. The notation looks like:
$\text{If}\phantom{\rule{8px}{0ex}}r\left(x\right)=\left(m\phantom{\rule{8px}{0ex}}\circ \phantom{\rule{8px}{0ex}}n\phantom{\rule{8px}{0ex}}\circ \phantom{\rule{8px}{0ex}}o\phantom{\rule{8px}{0ex}}\right)x=m\left\{n\left[o\left(x\right)\right]\right\}$
THEN
$r\text{'}\left(x\right)=m\text{'}\left\{n\left[o\left(x\right)\right]\right\}·n\text{'}\left[o\left(x\right)\right]·o\text{'}\left(x\right)$
Here, we look for the derivative of the compound function:
$f\left(x\right)={sin}^{3}\left(3x-2\right)$
Chain Rule - Three or More Functions