So, basically, you want to find d/dx(arctan(x^2y)).
We need to first observe that y and x have no relation to each other in the expression. This observation is very important, since now y can be treated as a constant with respect to x.
We first apply chain rule:
d/dx(arctan(x^2y)) = d/(d(x^2y))(arctan(x^2y)) xx d/dx(x^2y) = 1/(1 + (x^2y)^2) xx d/dx(x^2y).
Here, as we mentioned earlier, y is a constant with respect to x. So,
d/dx(x^2 color(red)(y)) = color(red)(y) xx d/dx(x^2) = 2xy
So, d/dx(arctan(x^2y)) = 1/(1 + (x^2y)^2) xx 2xy = (2xy)/(1 + (x^2y)^2)