If you're like me and don't remember hyperbolic identities (especially involving inverse functions) off the top of your head, recall the definitions of the hyperbolic cosine and sine:
Then differentiating yields
so that by the chain rule, if
then
Now, let , so that (•) .
Recall that
and so the derivative of tanh(<em>x</em>) is
where the last equality follows from the hyperbolic Pythagorean identity,
Differentiating both sides of (•) implicitly with respect to <em>x</em> gives
So, the derivative we want is the somewhat messy expression
and while this could be simplified into a rational expression of <em>x</em>, I would argue for leaving the solution in this form considering how <em>y</em> is given in this form from the start.
In case you are interested, we have
and you can instead work on differentiating that; you would end up with