Final answer:
To find dy/dx of X+3xy+y^2=2x, we differentiate both sides with respect to x using implicit differentiation and solve for dy/dx to get the final result of (1 - 3y) / (3x + 2y).
Step-by-step explanation:
To find dy/dx for the given equation X+3xy+y^2=2x, we need to use implicit differentiation since both x and y are intermingled. We differentiate both sides of the equation with respect to x, keeping in mind to apply the product rule to the term 3xy and the chain rule to y^2.
Differentiating term by term, the derivative of x is 1, for 3xy it's 3(y + x(dy/dx)), and for y^2 it's 2y(dy/dx). The derivative of 2x is simply 2. Putting it all together, we get:
1 + 3(dy/dx)x + 3y + 2y(dy/dx) = 2
Now, collecting all dy/dx terms on one side and the rest on the other side, we have:
(3x + 2y) dy/dx = 2 - 1 - 3y
dy/dx = (2 - 1 - 3y) / (3x + 2y)