Differentiating inverse hyperbolic function

Question

I am trying to differentiate $\tanh^{−1}\left(x/(1 + x^2)\right)$, but am finding it difficult understanding what to do. I think you have to place the differential of the angle of the hyperbolic function as the numerator so i differentiated it and got $(1 - x^2)/(1 + x^2)^2$. But i am stuck about what to do next.

Answer 1

First it might help to have on hand that $$[\text{arctanh}(y)]^\prime = \frac{1}{1-y^2}\cdot y^\prime$$ Now just plug in $$y = \frac{x}{1+x^2}$$ and the result you have for $y'$ to get your answer.