Final answer:
The partial derivative of x with respect to z, for the equation 3xy - z = 3, is found by implicit differentiation, holding y constant. After differentiation, we get ∂x/∂z = 1/(3y).
Step-by-step explanation:
The student is asking for the partial derivative of x with respect to z, given the equation 3xy - z = 3. To find the partial derivative ∂x/∂z, we need to treat y as a constant and solve for ∂x/∂z implicitly. This involves differentiating both sides of the equation with respect to z, while keeping y constant, and then solving for ∂x/∂z.
Let's differentiate the given equation:
- ∂/∂z(3xy) - ∂/∂z(z) = ∂/∂z(3)
- 3y(∂x/∂z) - 1 = 0
- 3y(∂x/∂z) = 1
- ∂x/∂z = 1/(3y)
Therefore, the partial derivative of x with respect to z, given the equation 3xy - z = 3, is 1/(3y).