Final answer:
To find the partial derivatives of z = u * v - w, we differentiate with respect to each variable treating the others as constants. The partial derivative with respect to u is v, with respect to v is u, and with respect to w is -1.
Step-by-step explanation:
To find the indicated partial derivatives of the function z = u * v - w with respect to the variables u, v, and w, we will use the rules of partial differentiation where each derivative is taken with respect to one variable, treating all other variables as constants.
Partial derivative with respect to u (∂z/∂u):
Holding v and w constant, the derivative of u * v with respect to u is just v, and the derivative of w with respect to u is zero since w is treated as a constant.
∂z/∂u = v
Partial derivative with respect to v (∂z/∂v):
Similar to the previous step, hold u and w constant, and differentiate u * v with respect to v, which is just u.
∂z/∂v = u
Partial derivative with respect to w (∂z/∂w):
Since u * v does not depend on w, its derivative with respect to w is zero, and the derivative of -w with respect to w is -1.
∂z/∂w = -1