Question

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

Answer 1

`3tan^2 x sec^2 x + 3x^2 sec^2x^3`

Explanation:

Power rule: `(n^n)' = n u^(n-1) u'`
`(tan u)' = u' sec^2 u`

Let `n = 3`

Rewrite original equation: `f(x) = (tan x)^3 + tan(x^3)`

Use the chain rule:
`f(x)' = 3(tan x)^2 (sec^2 x) + sec^2 (x^3) (3x^2)`

Rearrange: `f(x)' = 3tan^2 x sec^2 x + 3x^2 sec^2x^3`