If the problem is,

Differentiate the following with respect to x.

a) xy^2.

Does it mean,

a) y=xy^2

dy/dx = 2yx(dy/dx) + y^2

so dy/dx=y^2/(1-2yx)

b) xy^2=0

2yx(dy/dx) + y^2 = 0

dy/dx=y^2/2yx

or

c) who knows so no assumption should be made

2yx(dy/dx) + Y^2

Thanks.