Hello, so I have this problem that I am not completely sure on, here it is:

Let f be the real-valued function defined by $\displaystyle f(x)=sin^3x + sin^3|x|$

(a) Find $\displaystyle f'(x) for x>0.$

(b) Find $\displaystyle f'(x) for x<0.$

(c) Determine whether $\displaystyle f(x)$ is continuous at $\displaystyle x=0$. Justify your answer

(d) Determine whether the derivative of $\displaystyle f(x)$ exists at $\displaystyle x=0$. Justify your answer.

So I know how to do power rule and I have memorized some standards derivative values. But I do not know about raising sin to the third power. If $\displaystyle x$ was cubed I would be able to do it, but not sin.

For b it would be different because the absolute value of x $\displaystyle = \sqrt(x^2)$. I do not know how to take the derivative of that.

For c I said no because sin is never continuous at 0?

Finally for d i out that it does not because $\displaystyle f(x)$ is not continuous at 0.

ANY help would be greatly appreciated once again. Thanks.