I want to determine all the points where $g(x) = |\sin(2x)|$ is differentiable.
A function is differentiable at a point if the left and right limits exist and are equal.
So it follows that $g(x)$ is differentiable for all $x$ except where $g(x) = 0$. For example, the derivative of $|\sin(2x)|$ does not exist at $x=0$.
Is this correct?
Best Answer
It is important to make the distinction between the limit of the function and the limit of the difference quotient.
Suppose you were simply talking about the limit of the function. We have $$\lim_{x\to0}g(x)=\lim_{x\to0}|\sin2x|=0.$$
However, if you were talking about the difference quotient, the limit would not exist: $$\lim_{h\to 0^-}\frac{|\sin2(x+h)|-|\sin2x|}{h}=-2,\quad \lim_{h\to 0^+}\frac{|\sin2(x+h)|-|\sin2x|}{h}=2.$$