To find the derivative b(dy/dx) from the equation 8y cos(x) + x^2 y^2 = 0, we need to use implicit differentiation.
1. Start by differentiating both sides of the equation with respect to x:
8(dy/dx) cos(x) – 8y sin(x) + 2xy^2 + x^2(2y)(dy/dx) = 0
2. Now, group the terms that contain dy/dx:
8(dy/dx) cos(x) + 2xy^2(dy/dx) = 8y sin(x) – 2xy^2
3. Factor out dy/dx from the left side:
dy/dx(8 cos(x) + 2xy^2) = 8y sin(x) – 2xy^2
4. Finally, solve for dy/dx:
dy/dx = rac{8y sin(x) – 2xy^2}{8 cos(x) + 2xy^2}
This gives us the rate of change of y with respect to x for the given equation. Make sure to substitute in any specific values for x or y if needed for particular cases!