Final answer:
To find d^2y / dx^2 at the point (-1, -2), differentiate the equation with respect to x twice, substitute the given coordinates, and solve for the second derivative. The value of d^2y / dx^2 at (-1, -2) is -10.
Step-by-step explanation:
To find d^2y / dx^2 at the point (-1, -2), we need to take the second derivative of y with respect to x and then evaluate it at x=-1 and y=-2. Let's start by differentiating the given equation with respect to x to find dy/dx.
Given: y+2x^3=-y^2y+2x
Differentiating both sides with respect to x:
2x^2 + (dy/dx) = -2y*(dy/dx) + 2
Next, we differentiate again with respect to x to find d^2y / dx^2.
2 + (d^2y / dx^2) = -2*(dy/dx)^2
Now, let's substitute the coordinates (-1, -2) to find d^2y / dx^2 at that point.
2 + (d^2y / dx^2) = -2*(-2)^2
2 + (d^2y / dx^2) = -8
(d^2y / dx^2) = -10