section 13.5 chain rule
DESCRIPTION
Section 13.5 Chain Rule. Calculus III November 16, 2009 Berkley High School [email protected]. Remembering Single Variable. Total Differential. On To Multi-Variable. Even more complicated…. Implicit Differentiation. F. y. x. x. x. Implicit Differentiation. Assignment. - PowerPoint PPT PresentationTRANSCRIPT
2
Remembering Single Variable
If
Then
h x f g x
h x f g x g x
If
and
Then
y f x x g t
dy dy dx
dt dx dt
3
Total Differential
If
,
Then
z f x y
z zdz dx dy
x y
4
On To Multi-Variable
If
, and and
Then
z f x y x g t y h t
dz z dx z dy
dt x dt y dt
z
x y
tt
z
x
z
y
dy
dt
dx
dt
5
2 2
2
2 2
0 2 0 0
, where sin and .
Find when 0.
2 , cos , 2 ,
2 cos 2 2sin cos sin 2
at 0, 2 sin 0 cos 0 sin 0 2 2
t
t
t t t t
w x y y x t y e
dwt
dtdw w dx w dy
dt x dt y dt
w dx w dyxy t x y e
x dt y dt
dwxy t x y e t e t t e e
dtdw
t e e edt
w
x y
tt
w
x
w
y
dy
dt
dx
dt
6
Even more complicated…
,
,
,
z f x y
x g s t
y h s t
z z x z y
s x s y s
z z x z y
t x t y t
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2 2
2 2 22 2
2 , where and .
Find
12 , 2 , 2 ,
1 1 4 2 62 2 2 2 2 2 2 2
sz xy x s t y
tz
sz z x z y
s x s y s
z x z yy s x
x s y s t
z s s s sy s x s s t t t
s t t t t t t
z
x y
t
z
x
z
y
x
t
x
s
s t
y
t
y
s
s
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2 2
2
2 22 2
3 3
2 2
2 , where and .
Find
2 , 2 , 2 ,
2 2 2 4 2
2 24 2 2
sz xy x s t y
tz
tz z x z y
t x t y t
z x z y sy t x
x t y t t
w s s sy t x t s t
t t t t
s ss s s
t t
z
x y
t
z
x
z
y
x
t
x
s
s t
y
t
y
s
s
9
Implicit Differentiation
( , ), , 0
( , ) 0
0
F x y F x f x
d dF x y
dx dxF dx F dy
x dx y dx
F dy F
y dx x
x
y
FFdy x
Fdx Fy
F
x y
xx
F
x
F
y
dy
dx
dx
dx
10
3 2 2
3 2 2
2
2 2
5 4 0
Let ( , ) 5 4
2
3 2 5
2 2
3 2 5 3 2 5
x
y
x
y
y y y x
F x y y y y x
F x
F y y
Fdy x x
dx F y y y y
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Implicit Differentiation
( , , ), , , , 0
, yx
z z
F x y z F x y f x y
FFz z
x F y F
12
Assignment
13.5, 1-9 odd, 15-37 odd