FIND dy/dx by implicit differentiation1.x^3+y^3-6xy=02.x^2+xy^2+y^2=13.sqrt(x+y)+xy=214.(x-y)^3=(x+y)^25.y=4(x^2+y^2)6.y^2=(3x+1/2x-3)7.y^2-3x+2y=0

Sure, let's solve these one by one:

1. x^3 + y^3 - 6xy = 0 Differentiating with respect to x, we get: 3x^2 + 3y^2*(dy/dx) - 6y - 6x*(dy/dx) = 0 Solving for dy/dx, we get: dy/dx = (6y - 3x^2) / (6x - 3y^2)

2. x^2 + xy^2 + y^2 = 1 Differentiating with respect to x, we get: 2x + y^2 + 2xy*(dy/dx) + 2y*(dy/dx) = 0 Solving for dy/dx, we get: dy/dx = (-2x - y^2) / (2xy + 2y)

3. sqrt(x+y) + xy = 2 Differentiating with respect to x, we get: 1/2*(1/sqrt(x+y)) + y + x*(dy/dx) = 0 Solving for dy/dx, we get: dy/dx = (-1/2*(1/sqrt(x+y)) - y) / x

4. (x-y)^3 = (x+y)^2 Differentiating with respect to x, we get: 3*(x-y)^2*(1 - dy/dx) = 2*(x+y)(1 + dy/dx) Solving for dy/dx, we get: dy/dx = (3(x-y)^2 - 2*(x+y)) / (3*(x-y)^2 + 2*(x+y))

5. y = 4(x^2 + y^2) Differentiating with respect to x, we get: dy/dx = 4*(2x + 2y*(dy/dx)) Solving for dy/dx, we get: dy/dx = 8x / (4 - 8y)

6. y^2 = (3x + 1/2x - 3) Differentiating with respect to x, we get: 2y*(dy/dx) = 3 - 1/(2x^2) Solving for dy/dx, we get: dy/dx = (3 - 1/(2x^2)) / (2y)

7. y^2 - 3x + 2y = 0 Differentiating with respect to x, we get: 2y*(dy/dx) - 3 + 2*(dy/dx) = 0 Solving for dy/dx, we get: dy/dx = 3 / (2y + 2)