\(f(x)=\arctan x\)
\(\dfrac{\mathrm df}{\mathrm dx}=\dfrac1{1+x^2}\)
\(\dfrac{\mathrm d^2f}{\mathrm dx^2}=-\dfrac{2x}{(1+x^2)^2}\)
\(\dfrac{\mathrm d^3f}{\mathrm dx^3}=\dfrac{6x^2-2}{(1+x^2)^3}\)
\(\dfrac{\mathrm d^4f}{\mathrm dx^4}=\dfrac{24x-24x^3}{(1+x^2)^4}\)
The Taylor polynomial takes the form
\(P_4(x)=f(0)+\dfrac{f'(0)}{1!}(x-0)+\dfrac{f''(0)}{2!}(x-0)^2+\dfrac{f'''(0)}{3!}(x-0)^3+\dfrac{f^{(4)}(0)}{4!}(x-0)^4\)
\(P_4(x)=\dfrac11x-\dfrac26x^3=x-\dfrac13x^3\)
Notice that the fourth degree term vanishes, since \(f^{(4)}(0)=0\), so you can stop at the third degree.
