分数関数1/(x+1)の導関数

$$\begin{aligned} \hspace{10pt}y' & =\lim_{h \to 0}\frac{1}{h}\left(\frac{1}{x+h+1}-\frac{1}{x+1}\right)\hspace{10pt}\\[0.5em] & = \lim_{h \to 0}\frac{1}{h}\frac{x+1-(x+h+1)}{(x+h+1)(x+1)}\\[0.5em] & = \lim_{h \to 0}\frac{1}{h}\frac{-h}{(x+h+1)(x+1)}\\[0.5em] & = \lim_{h \to 0}\frac{-1}{(x+h+1)(x+1)}\\[0.5em] & = -\frac{1}{(x+1)^2}\\[0.5em] \end{aligned}$$