So the first thing to do is notice that this is a fraction and the quotient rule must be applied. The general form of the quotient rule is d/dy[f(y)/g(y)] = (g(y)f'(y)-g'(y)f(y))/(g(y))^2.
Time to assign f(y) and g(y) and calculate their derivatives f'(y) and g'(y). The top function is our f(y) so f(y)=7y+y^2 and g(y)=y+1. Their derivatives are f'(y)=7+2y and g'(y)=1. Now that we have each separate piece of the derivative, it's time to plug them into the whole.
((1+y)(7+2y)-(7y+y^2)(1))/(1+y)^2. Then you distribute the factors on top and combine like terms to get the answer above.