To find dy/dx, we need to use implicit differentiation.
Given the equation: x^2y^2 = sin(n + y)
Step 1: Differentiate both sides of the equation with respect to x.
d/dx(x^2y^2) = d/dx(sin(n + y))
Step 2: Apply the product rule on the left side of the equation.
2x * y^2 + x^2 * 2y(dy/dx) = cos(n + y) * (d/dx(n + y))
Step 3: Simplify the equation.
2xy^2 + 2x^2y(dy/dx) = cos(n + y) * (0 + dy/dx)
Step 4: Solve for dy/dx.
2x^2y(dy/dx) = cos(n + y) * dy/dx - 2xy^2
Step 5: Factor out dy/dx from the right side of the equation.
2x^2y(dy/dx) - cos(n + y) * dy/dx = -2xy^2
Step 6: Factor out dy/dx on the left side of the equation.
dy/dx(2x^2y - cos(n + y)) = -2xy^2
Step 7: Solve for dy/dx.
dy/dx = -2xy^2 / (2x^2y - cos(n + y))
So, the derivative dy/dx of the given equation is -2xy^2 / (2x^2y - cos(n + y)).