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Theorem 5.21
Let
f be a function of two variables (
x,
y), differentiable on an open domain
.
Suppose that
x and
y are functions of two independent
variables
u,
v, differentiable on an open domain
,
and such that for every
.
Then
w=f ( x(u,v),y(u,v)) is a function of u,v, differentiable on
and we have:
Figure 3:
The chain rule for two intermediate variables.

Example 5.22
Let
w=
f(
x,
y)=
xy, where
x=
u+
v and
y=
u
v. Then
w=(
u+
v)(
u
v)=
u^{2}
v^{2} and we have:
By the chain rule, we have:
Theorem 5.23
Let
f be a function of three variables (
x,
y,
z), differentiable on an open domain
.
Suppose that
x,
y and
z are functions of two independent
variables
u,
v, differentiable on an open domain
,
and such that for every
.
Then
w=f ( x(u,v),y(u,v),z(u,v) ) is a function of u,v, differentiable on
and we have:
Remark 5.24
Actually
(x(u,v),y(u,v), z(u,v)) is a parametrization of a surface in the 3dimenasional space. The chain rule enables us to compute the derivative of a function f defined on a surface.
Next: Implicit differentiation.
Up: The chain rule.
Previous: One independent variable.
Noah DanaPicard
20010530