Finding dx/dy where x3+y3=6xy

dx/dy=2x-y^2/x^2-2y

Cinar is correct, but are you sure you don't want dy/dx?

If so, just take the reciprocal.

To find dx/dy, we need to differentiate both sides of the equation x^3 + y^3 = 6xy with respect to y.

Differentiating x^3 with respect to y:

d/dy(x^3) = 3x^2 * dx/dy

Differentiating y^3 with respect to y:

d/dy(y^3) = 3y^2 * dy/dy

Since dy/dy is simply equal to 1, we can simplify it to:

d/dy(y^3) = 3y^2

Differentiating 6xy with respect to y:

d/dy(6xy) = 6x * dy/dy

Again, since dy/dy is equal to 1, we can simplify it to:

d/dy(6xy) = 6x

Now, let's substitute these derivatives back into the original equation:

3x^2 * dx/dy + 3y^2 = 6x

Rearranging the equation:

3x^2 * dx/dy = 6x - 3y^2

Dividing both sides by 3x^2:

dx/dy = (6x - 3y^2) / (3x^2)

Simplifying the equation further:

dx/dy = 2 - (y^2 / x^2)

So, the derivative dx/dy of x^3 + y^3 = 6xy is given by the equation dx/dy = 2 - (y^2 / x^2).