Gradient Quesiton

Question

So I was struggling with finding the gradientto this problem

$$f(x,y)= \frac{Ax + By}{Cx + Dy}$$

Down below is my work, but I apparently did something wrong. Where did I mess up?

Answer 1

You didn't mess up, the working is fine.

Note that you can factorize $\frac{AD-BC}{(Cx+Dy)^2}$ out from both component and just write them as $$\frac{AD-BC}{(Cx+Dy)^2}(y \vec{i}-x\vec{j})$$