Final answer:
To solve for dy/dx in the equation 6y cos(x) = x² y², apply implicit differentiation, rearrange the terms, and solve for dy/dx. The result is dy/dx = (2xy² + 6y sin(x)) / (6 cos(x) - 2x² y).
Step-by-step explanation:
To find dy/dx, the derivative of y with respect to x, we need to utilize implicit differentiation on the equation 6y cos(x) = x² y². First, we'll differentiate both sides of the equation with respect to x:
After differentiation, we have: 6dy/dx cos(x) - 6y sin(x) = 2xy² + x² 2y(dy/dx). Now we need to solve for dy/dx. We'll move all terms involving dy/dx to one side of the equation and factor dy/dx out:
dy/dx (6 cos(x) - 2x² y) = 2xy² + 6y sin(x)
Thus, dy/dx = (2xy² + 6y sin(x)) / (6 cos(x) - 2x² y).
This final expression represents the derivative dy/dx for the given equation.