The
Chain Rule for Functions of Several Variables
(continued)
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So the two partial derivatives for w(x,y) are as
follows: (note “product” of partials) ∂w/∂x =
[∂w/ ∂u] ∂u/∂x +
[∂w/ ∂v] ∂v/∂x and ∂w/∂y =
[∂w/ ∂u] ∂u/∂y +
[∂w/ ∂v] ∂v/∂y |
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Now consider the case
where the dependent variable, w, has three intermediate variables and two
independent variables. w =
w(x, y, z) where x
= x(u,v), y = y(u,v), and z
= z(u,v) w is the dependent variable, x,
y, and z
are intermediate variables and u
and v are the independent variables (note
“product” of partials) So ∂w/∂u =
[∂w/ ∂x] ∂x/∂u +
[∂w/ ∂y] ∂y/∂u +
[∂w/ ∂z] ∂z/∂u and ∂w/∂v =
[∂w/ ∂x] ∂x/∂v +
[∂w/ ∂y] ∂y/∂v +
[∂w/ ∂z] ∂z/∂v |
Copyright © 2017 Richard C. Coddington
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