$f(x)=\dfrac{2}{x+1}$
$\to \Delta y=f(x_o+\Delta x)-f(x_o)=\dfrac{2}{2+\Delta x+1}-\dfrac{2}{2+1}$
$=\dfrac{2}{\Delta x+3}-\dfrac{2}{3}$
$=\dfrac{2.3-2\Delta x-2.3}{3(\Delta x+3)}$
$=\dfrac{ -2\Delta x}{3\Delta x+9}$
$\to \dfrac{\Delta y}{\Delta x}=\dfrac{ -2}{3\Delta x+9}$