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).