Final answer:
To find dy/dz, differentiate both sides of the equation with respect to z. dy/dz = 0. For dy/dx, differentiate both sides of the equation with respect to x using the product rule.
Step-by-step explanation:
To find dy/dz for the equation 3x³y² - 2x²y = 6, we need to differentiate both sides of the equation with respect to z. Since there is no 'z' term in the equation, the derivative of any term with respect to z would be zero. Therefore, dy/dz = 0.
To find dy/dx, we differentiate both sides of the equation with respect to x. Using the product rule, the derivative of the first term is 9x²y² and the derivative of the second term is -4xy. Setting this equal to zero and solving for y, we get y = 0 or y = 4x/9. Therefore, dy/dx = 0 or dy/dx = 4x/9.