Implicit differentiation?

Question

y=sin (xy^2)

Only the y in the parentheses is squared...not the x. You don't have to show your work. Just give me an answer. Thanks.

But when you are differentiating the left side, doesn't it equal y (dy/dx) ? Or is it just dy/dx ?

Answer 1

y´= cos(xy^2)*(y^2+2xy*y´)so
y´[1-2xy cos(xy^2)]= y^2*cos(xy^2)
so
y´= y^2 cos(xy^2)/[1-2xy*cos(xy^2)]

Answer 2

dy/dx
= (cosxy^2) [y^2 + 2xy dy/dx]

(1- 2xy cosxy^2) dy/dx = y^2 cosxy^2

dy/dx = (y^2 cosxy^2) / (1- 2xy cosxy^2)