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)