Differentiate artanh(x) with respect to x
First we set y=artanh(x). Then we rearrange such that tanh(y)=x. There several approaches to find dy/dx, but the quickest is to use implicit differentiation.
The differential of tanh(y) is sech2y. We differentiate both sides with respect to x using implicit differentiation so that tanh(y)=x becomes sech2(y)(dy/dx)=1. We now rearrange this:
dy/dx=1/sech2y
We use the identity sech2y=1-tanh2y , and since x=tanh(y), we have
dy/dx=1/(1-tanh2y)= 1/(1-x2)
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