the derivative of: $f(x)= x-2\text{arctan}(x)$

Question

Now, I know that this will just become $1-\frac{2}{x^2+1}$ if I apply the derivative of arctan, but how can I calculate the derivative of this function, step by step? I am already lost at the beginning. If $f(x)=\text{arctan}(x)$ instead I would just substitute it so that $\text{tan}(y) = x$ and then use implicit differentiation, but I don't know where to start with the function in the title.

Answer 1

If you really want to apply this method then we would have $$y=x-2\arctan{(x)}$$ $$x=\tan{\left(\frac{x-y}2\right)}$$ Which we can differentiate both sides with respect to $x$ giving $$1=\sec^2{\left(\frac{x-y}2\right)}\cdot\left(\frac{1-y'}2\right)$$ $$1=(1+x^2)\cdot\left(\frac{1-y'}2\right)$$ $$\frac{1-y'}2=\frac1{1+x^2}$$ $$1-y'=\frac2{1+x^2}$$ $$y'=1-\frac2{1+x^2}$$