Final answer:
To prove that the product of ∂z/∂x and ∂z/∂y is zero for z = f(x - y), we used the chain rule to find that one partial derivative is the negation of the other, making their product zero.
Step-by-step explanation:
The student has asked to show that if z = f(x - y), then the product of the partial derivatives ∂z/∂x and ∂z/∂y is zero, which is ∂z/∂x ∂z/∂y = 0. Assuming z is a function of the single variable u = x - y, ∂z/∂x = f'(u) and ∂z/∂y = -f'(u) due to the chain rule. Since the derivatives with respect to x and y have the same magnitude but opposite signs, their product is indeed zero, which confirms the statement.