12.4 Chain Rule

12.4 Chain Rule - 1 2 .4 
C h a in 
R u l e 
 


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Unformatted text preview: 1 2 .4 
C h a in 
R u l e 
 
 Suppose
z
=
f(x,y),
x
=
g(t),
and
y
=
h(t).

Ultimately
z
depends
 dz on
t.

We
want
to
find
 dt .
 
 df Ex.

f(x,
y)
=
x2y,

x
=
3t
+
1,
y
=
6t2
:










Find
 dt .
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex.

f(x,
y)
=
ln(xy),

x
=
et,

y
=
e‐t
:












Find
 df dt .
 
 
 
 
 
 
 
 
 Ex.



fx
=
cost,

fy
=
sint,

x
=
t2
–
4t,


y
=
lnt
:
 df Find
 dt .
 
 
 Suppose
z
=
f(x,
y),
x
=
g(s,
t),
and
y
=
h(s,
t).

Then
z
ultimately
 depends
on
s
and
t.
 
 
 
 
 
 u z = x e , x = , y = v ln u : Ex.
 v 2y ∂z Find ∂v 
 Evaluate
the
partial
derivative
at
(u,
v)
=
(1,
2).
 
 
 
 
 
 
 
 
 Ex.

Suppose
z
=
f(x,
y),

x
=
g(s,
t),
and
y
=
h(s,
t)
 fx(2,
1)
=
4,

fy(2,
1)
=
6,



gs(‐1,
3)
=
3,


hs(
‐1,
3)
=
‐5,
 g(‐1,
3)
=
2,

and
h(‐1,
3)
=
1
 Find
fs(2,
1).
 
 
 
 
 
 
 
 Ex.

Suppose
w
=
f(r,
s,
t),

r
=
g(x,
y),

s
=
h(x,
y),
and
t
=
k(x,
y)
 
 
 
 
 
 
 dz dz and Ex.

Suppose
yz
=
ln(x
+
z).



Find
 dx dy .
 dz F =− x, dx Fz 
 
 
 
 
 
 
 
 
 Fy dz =− dy Fz 
 u dz 2y z = x e , x = , y = v ln u .
 Do:

1.

Find
 du 

for
 v 













Evaluate
it
for
(u,
v)
=
(1,
2).
 ⎛ ∂z ⎞ ⎛ ∂z ⎞ 








2.

If
z
=
f(x,
y)
where
x
=
s
+
t
and
y
=
s
–
t,

find
 ⎝ ∂s ⎠ ⎝ ∂t ⎠ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Applications
 Ex.
The
radius
of
a
right
circular
cylinder
is
increasing
at
a
rate
 of
2
in/min
and
the
height
is
increasing
at
3
in/min.

How
fast
 is
the
lateral
surface
area
changing
when
the
radius
is
10
 inches
and
the
height
is
12
inches?
 
 
 
 
 
 
 
 Ex.

The
lengths
x,
y,
z
of
the
edges
of
a
rectangular
box
are
 changing
with
time.

At
the
instant
in
question,
V
=
6
m3,

 dx dy dV = = 1 m / s and = 9 m 3 / s.
 x
=
1m,
y
=
2
m,
 dt dt dt How
fast
is
z
changing
at
that
instant?
 
 
 
 ...
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This note was uploaded on 10/02/2011 for the course AERO 1234 at Virginia Tech.

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