Question

Chain Rule: Two and Three Independent Variables In Exercises 7 and 8, (a) express ∂z/∂u and ∂z/∂v as functions of u and v both by using the

Answer 1

To express ∂z/∂u and ∂z/∂v as functions of u and v using the chain rule, we need to consider the partial derivatives of z with respect to each independent variable in terms of u and v.

Step-by-step explanation:

To express ∂z/∂u and ∂z/∂v as functions of u and v using the chain rule, we need to consider the partial derivatives of z with respect to each independent variable in terms of u and v. The chain rule states that if z is a function of u and v, and u and v are themselves functions of some other variable, say x, then the partial derivatives of z with respect to u and v can be expressed in terms of the partial derivatives of u and v with respect to x, and the partial derivatives of z with respect to x.

We can express the partial derivative ∂z/∂u as:

∂z/∂u = (∂z/∂x)(∂x/∂u) + (∂z/∂y)(∂y/∂u)

Similarly, we can express the partial derivative ∂z/∂v as:

∂z/∂v = (∂z/∂x)(∂x/∂v) + (∂z/∂y)(∂y/∂v).