Question

Find dy/dx.
x^2y+xy^2=6

Answer 1

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.

Answer 2

we are asked to evaluate dx/dy of the function x^2y+xy^2=6
we use implicit differentiation here:
2xy dx + 2xy dy = 0we can cancel 2xy from both terms in the left-hand side such that what is left is dx/dy = 0