Find dy/dx by implicit differentiation.

Question

arctan(2x^2y)=x+4xy^2

Answer 1

arctan(2x^2y)=x+4xy^2

Using the good old product and chain rules, we get

1/(1+(2x^2y)^2) * (4xy + 2x^2y') = 1 + 4y^2 + 8xyy'

Now just collect terms and solve for y':

16x^4y^4 + 4x^4y^2 + 4y^2 - 4xy + 1
- -------------------------------------------
2x(16x^4y^3 - x + 4y)