Determining differentiability of $h(x)= [\sin^3 x ]+\{\sin^{1/3}x\}$

Question

$h(x) = [\sin^3 x ]+\{\sin^{1/3}x\}$ for all $x \in (4n+1)\dfrac \pi 2 , n \in \mathbb{Z}$

[x], {x} denote floor and fractional part functions respectively.

How do I determine the differentiability of $h(x)$?

Attempt:

$f(x+) = f(x-)= f(x)= 1$

$\implies $ The function is continuous.

The problem I am facing is that I do not know how to find the derivative of fractional and floor functions.

Answer 1

Note that the function $$h(x) = \lfloor\sin^3 x\rfloor+\{\sin^{1/3}x\},$$ $x=\{n\in Z:2n+\dfrac\pi2 \}$ is not continuous (being defined only for an isolated set of points).

Thus $h(x)$ is not differentiable.