TITLE: Finding derivative of $\sqrt[3]{\sin(2x)}$ using only definition of derivative
QUESTION [10 upvotes]: First post here, so hello everyone.
Here's the problem:

Find the first derivative of: $$\sqrt[3]{\sin(2x)}$$ 

But, you can only use the difference quotient... 
(i.e. the limit of $\frac{f(x+h)-f(x)}{h}$ as $h \rightarrow 0$)
This is a particular problem given by my prof. It's not for credit or even an assignment, she gave it as an interestingly difficult problem with some twist and turns. I've worked on it for several hours now and would like to see how others approach solving it as I'm wrapped around an axel after using double-angle ID and clearing the cubes in the numerator. Any advice is appreciated.
Thanks.

REPLY [2 votes]: Hint:
Use symmetry! :
$$\sin(2x+2h)=\sin((2x+h)+h) \quad \sin(2x)=\sin((2x+h)-h)$$