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Theorem 5.16
Let
f be a function of two variables (
x,
y), differentiable on an open domain
.
Suppose that
x and
y are functions of a single variable
t, differentiable on an open interval
,
and such that for every
.
Then
f ( x(t),y(t) ) is a function of t, differentiable on
and we have:
Figure 1:
The chain rule for two intermediate variables.

Remark 5.17
Actually
(x(t), y(t)) is a parametrization of a curve in the plane. The chain rule enables us to compute the derivative of a function f defined on a plane curve.
Example 5.18
Let
z=
f(
x,
y)=
x^{2}+
y^{3}. Then
and
.
Suppose now that
and .
Then
and
.
It follows:
A direct computation shows that, as
,
the derivative
is equal to
.
Theorem 5.19
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 a single variable
t, differentiable on an open interval
,
and such that for every
.
Then
w=f ( x(t),y(t),z(t) ) is a function of t, differentiable on
and we have:
Figure 2:
The chain rule for three intermediate variables.

Remark 5.20
Actually
(x(t), y(t), z(t)) is a parametrization of a curve in the 3dimensional space. The chain rule enables us to compute the derivative of a function f defined on a
space curve.
Next: Two independent variables.
Up: The chain rule.
Previous: The chain rule.
Noah DanaPicard
20010530