Derivative of absolute value function

Question

What is $f'(x)$ and $f''(x)$ of $f(x) = x^{1/3}\vert 4-x \vert$?

Do you use two cases or can it be solved a different way?

Answer 1

Ask yourself: is there any point on this curve where it is not differentiable? If there is, then the derivative clearly has to be in two 'pieces', so you have to use two cases.

Answer 2

I think it is easier to break it into two cases.
Another way is to use the derivative of $|x|$, which is $x/|x|$, but not defined at $x=0$,
and the second derivative which is $0$ but not defined at $x=0$.

Answer 3

It might be easier to do it by cases, but you can also use:

$\vert y \vert = (y^2)^{1/2}$