# How do you differentiate arcsin(csc(1-1/x^3)) ) using the chain rule?

Nov 3, 2016

As a Real valued function this is not differentiable.

#### Explanation:

This is not differentiable as a Real valued function, since $\csc \theta \in \left(- \infty , - 1\right] \cup \left[1 , \infty\right)$ and $\sin \theta \in \left[- 1 , 1\right]$, so the only points at which $\arcsin \left(\csc \left(1 - \frac{1}{x} ^ 3\right)\right)$ is defined are the discrete points at which:

$1 - \frac{1}{x} ^ 3 = \frac{\left(2 k + 1\right) \pi}{2}$

That is:

$x \in \left\{\sqrt[3]{\frac{1}{1 - \frac{\left(2 k + 1\right) \pi}{2}}} : k \in \mathbb{Z}\right\}$

If you had asked about $\arcsin \left(\csc \left(1 - \frac{1}{z} ^ 3\right)\right)$ then I would look at Complex trigonometric functions.