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)#