Math 200, Spring 2010
Handout 14
Section 146
The Chain Rule
The Chain rule (Case 1)
Suppose that
(
)
,
z
f
x y
=
is a differentiable function of
x
and
y
, where
( )
x
x t
=
and
( )
y
y t
=
are both
differentiable function of
t
.
Then
z
is a differentiable function of
t
and
dz
f dx
f dy
dt
x dt
y dt
∂
∂
=
+
∂
∂
.
1.
Use the chain rule to find
dz dt
where
(
)
4
cos
4
,
5
,
1
z
x
y
x
t
y
t
=
+
=
=
2.
The radius of a right circular cone is increasing at a rate of 1.8 in/s while its height is decreasing at a
rate of 2.5 in/s.
At what rate is the volume of the cone changing when the radius is 120 in. and the
height is 140 in.
The Chain rule (Case 2)
Suppose that
(
)
,
z
f
x y
=
is a differentiable function of
x
and
y
,
where
(
)
,
x
x s t
=
and
(
)
,
y
y s t
=
are
differentiable function of
s
and
t
.
Then
z
is a differentiable function of
s
and
t
and
z
z
x
z
y
s
x
s
y
s
∂
∂
∂
∂
∂
=
+
∂
∂
∂
∂
∂
.
z
z
x
z
y
t
x
t
y
t
∂
∂
∂
∂
∂
=
+
∂
∂
∂
∂
∂
Tree Diagram
s
and
t
are
independent
variables,
x
and
y
are
intermediate
variables,
and
z
is the
dependent
variable.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 JAMESDCAMPBELL
 Calculus, Chain Rule, Derivative, The Chain Rule, differentiable function

Click to edit the document details