Final answer:
To find dy/dx of the equation x^2y + xy^2 = 6, we use implicit differentiation, rearrange terms to isolate dy/dx, and divide by the sum of the product of x and the derivative terms to get dy/dx = (-2xy - y^2) / (x^2 + 2xy).
Step-by-step explanation:
To find dy/dx, we use implicit differentiation because the equation x^2y + xy^2 = 6 mixes x and y together. Differentiating both sides with respect to x gives us:
d/dx (x^2y) + d/dx (xy^2) = d/dx (6)
This results in:
2xy + x^2(dy/dx) + y^2 + 2xy(dy/dx) = 0
Now, we need to solve for dy/dx. We combine like terms:
x^2(dy/dx) + 2xy(dy/dx) + 2xy + y^2 = 0
Gather all the dy/dx terms on one side:
x^2(dy/dx) + 2xy(dy/dx) = -2xy - y^2
Factor out dy/dx:
dy/dx (x^2 + 2xy) = -2xy - y^2
Finally, divide by (x^2 + 2xy) to isolate dy/dx:
dy/dx = (-2xy - y^2) / (x^2 + 2xy)
And that gives us the derivative dy/dx.