Question

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

Answer 1

As a Real valued function this is not differentiable.

Explanation:

This is not differentiable as a Real valued function, since `csc theta in (-oo, -1] uu [1, oo)` and `sin theta in [-1, 1]`, so the only points at which `arcsin(csc(1-1/x^3))` is defined are the discrete points at which:

`1-1/x^3 = ((2k+1)pi)/2`

That is:

`x in { root(3)(1/(1-((2k+1)pi)/2)) : k in ZZ }`

If you had asked about `arcsin(csc(1-1/z^3))` then I would look at Complex trigonometric functions.