What is the derivative of (sin|x|)^3?
-for x>0
-for x<0
I think it is 3sin|x|*(cos|x|) for the first and 3sin|x|*(-cos|x|) for the second but I'm not sure. . .
alternatively, using the definition of $\displaystyle |x|$, you could realize you have the function:
$\displaystyle
f(x) = \left \{ \begin{array}{ll} (\sin x)^3 & \text{ if } x > 0 \\ (\sin (-x))^3 & \text{ if } x < 0 \end{array} \right. $
$\displaystyle {\color{white}f(x)} = \left \{ \begin{array}{ll} (\sin x)^3 & \text{ if } x > 0 \\ -(\sin x)^3 & \text{ if } x < 0 \end{array} \right. $
now differentiate piece by piece