Question

What is the derivative of `tan^4(3x)`?

Answer 1

If is helpful (until you really get used to it) to remember and rewrite:
`tan^4(3x) = (tan(3x))^4`

This makes it clear that ultimately (at the last), this is a fourth power function. We'll need the power rule and the chain rule. (Some authors call this combination "The General Power Rule".) In fact we'll need the chain rule twice.

`d/(dx)((tan(3x))^4) = 4 (tan(3x))^3 [d/(dx)(tan(3x))]`

`=4tan^3(3x)[sec^2(3x) d/(dx)(3x)]=4tan^3(3x)[3sec^2(3x)]`

`d/(dx)(tan^4(3x))=12tan^3(3x) sec^2(3x)`