Use the definition of derivative as a limit to find the derivative of
$\displaystyle {x}/{(x+1)}$
show me all steps please, thank you!
$\displaystyle \frac{\frac{x+h}{x+h+1}-\frac{x}{x+1}}{h}=\frac{\frac{(x+h)(x+1)}{(x+h+1)( x+1)}-\frac{x(x+h+1)}{(x+1)(x+h+1)}}{h}$ $\displaystyle =\frac{\frac{x^2+hx+x+h-(x^2+hx+x)}{(x+1)(x+h+1)}}{h}=\frac{\frac{h}{(x+1) (x+h+1)}}{h}=\frac{1}{(x+1)(x+h+1)}$
Taking the limit as $\displaystyle h\to0$, we have:
$\displaystyle \lim_{h\to0}\frac{1}{(x+1)(x+h+1)}=\frac{1}{(x+1)^ 2}$