Final answer:
To find dy/dx in terms of x and y from the given equation x²+ xy+ y³=0, we use implicit differentiation and differentiate both sides of the equation with respect to x. The derivative of x² is 2x, the derivative of xy is x(dy/dx) + y, and the derivative of y³ is 3y²(dy/dx). Equating the derivatives to 0 gives us the equation 2x + x(dy/dx) + y + 3y²(dy/dx) = 0. Solving for dy/dx, we get dy/dx = (-2x - y)/(x + 3y²).
Step-by-step explanation:
To find dy/dx in terms of x and y from the given equation x²+ xy+ y³=0, we need to differentiate both sides of the equation with respect to x. To do this, we'll use implicit differentiation.
Differentiating x² gives 2x. Differentiating xy gives x(dy/dx) + y. Differentiating y³ gives 3y²(dy/dx).
Equating the derivatives to 0 gives us the equation 2x + x(dy/dx) + y + 3y²(dy/dx) = 0. Solving for dy/dx, we get dy/dx = (-2x - y)/(x + 3y²).