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

1 Answer
George C.
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 (-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.

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