Here my question , for example x^2*y^2=0 when y=x=1, find d/dx, can I make the y^2 = x^2 and therefor change the equation to x^4=0 and differentiate for my x?
Yes, that is wrong. "At x= y= 1" does not mean that you can set x= y before taking the derivative. You have to take the derivative, with respect to y as (1, y) approaches (1, 1). So, because this is the partial derivative with respect to 1, you can set x= 1, making $\displaystyle f(x, y)= ln((1+2y)^2+ (2- y)^2+ 4y^2)$. Differentiate that and then set y= 1.