Find the derivative of the implicit function X^3+Y^3+6XY=0

Answer 1

To find the derivative of the implicit function x^3 + y^3 + 6xy = 0, we can use implicit differentiation.

Taking the derivative of each term with respect to x, we have:

d/dx (x^3) + d/dx (y^3) + d/dx (6xy) = d/dx (0)

3x^2 + 3y^2*(dy/dx) + 6x*(dy/dx) + 6y = 0

Rearranging the terms, we have:

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

Factoring out the common factor of dy/dx, we have:

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

Finally, we can solve for dy/dx:

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