Show that
Hmm, interesting. I actually don't agree with the result unless x is non-negative. If that was assumed in the problem statement, great. Otherwise, you can't simplify
You can say that So you could simplify as follows:
where sgn is the signum function.
Otherwise, your working is correct, I think.
Just present the proof you had in Post # 3, with the modifications mentioned in Post # 4 and Post # 4's answer. If the problem originally stated that x is positive, you can simplify the signum to the expression in the OP. Otherwise, I think you have to leave it in there. To be on the safe side, include a sentence explaining what you're doing.
You could always convert the inverse hyperbolic sine function to its logarithmic equivalent...
Of course, since for all , that means we can only accept .
Therefore
You can now find the derivative using the chain rule, or exponentiating both sides and using implicit differentiation.