Uppgift 2

Derivatans definition ger direkt att
$$\begin{aligned}f'(x)&=\lim_{h\to0}\frac{f(x+h)-f(x)}h=\lim_{h\to0}\frac{\frac1{(x+h)^2+1}-\frac1{x^2+1}}h\\&=\lim_{h\to0}\frac{(x^2+1)-((x+h)^2+1)}{h(x^2+1)((x+h)^2+1)}=\lim_{h\to0}\frac{-2x-h}{(x^2+1)((x+h)^2+1)}=\frac{-2x}{(x^2+1)^2}.\end{aligned}$$