So, basically, you want to find d/dx(arctan(x^2y))ddx(arctan(x2y)).
We need to first observe that yy and xx have no relation to each other in the expression. This observation is very important, since now yy can be treated as a constant with respect to xx.
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)ddx(arctan(x2y))=dd(x2y)(arctan(x2y))×ddx(x2y)=11+(x2y)2×ddx(x2y).
Here, as we mentioned earlier, yy is a constant with respect to xx. So,
d/dx(x^2 color(red)(y)) = color(red)(y) xx d/dx(x^2) = 2xyddx(x2y)=y×ddx(x2)=2xy
So, d/dx(arctan(x^2y)) = 1/(1 + (x^2y)^2) xx 2xy = (2xy)/(1 + (x^2y)^2)ddx(arctan(x2y))=11+(x2y)2×2xy=2xy1+(x2y)2