dy/dx=((x+1)-(x-1))/(x+1)^2=2/(x+1)^2. When x=0, dy/dx=2.
Another way of solving this is to examine the difference of (x+h-1)/(x+h+1)-(x-1)/(x+1);
((x+h-1)(x+1)-(x-1)(x+h+1))/((x+h+1)(x+1));
((x^2+hx-x+x+h-1)-(x^2+hx+x-x-h-1))/(x^2+hx+x+x+h+1);
(x^2+hx+h-1-x^2-hx+h+1)/(x^2+hx+2x+h+1);
2h/(x^2+2x+1+hx+h). When x=0 and h is very small this becomes: 2h/(h+1). h is small compared to 1 so if this expression shows the change in y for the small change in x given by h, the rate of change is 2h/(h(h+1))=2/(h+1). As h approaches zero this rate of change=2, the figure we obtained by calculus earlier.