Help with hyperbolic inverse trig Dif/integration

My course seems to have some gaps and I'm having a hard time understanding hyperbolic inverse trig differentiation.

differentiate tanh^{-1}(sinx)

Book says switch x and y which will make the inverse go away

so I get tanh(siny) and Dx both sides

Dx = sec^{2}h(siny)*Dx (sin y)

= sec^{2}h(sin y) * cosy and thats where I get stuck. None of this seems to be right.

similarly the next problem is sinh^{-1 }(tan x) Hopefully if I can understand the mechanics of the first I can work the 2nd one a little smoother.

Thanks

Re: Help with hyperbolic inverse trig Dif/integration

Let

$\displaystyle y = \tanh^{-1}(\sin x),$

then

$\displaystyle \tanh y = \sin x,$

and now differentiate implicitly.