2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.
-
Upload
aleesha-daniel -
Category
Documents
-
view
214 -
download
0
Transcript of 2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.
![Page 1: 2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.](https://reader035.fdocuments.us/reader035/viewer/2022070404/56649f335503460f94c50131/html5/thumbnails/1.jpg)
2.4 The Chain Rule
Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002
![Page 2: 2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.](https://reader035.fdocuments.us/reader035/viewer/2022070404/56649f335503460f94c50131/html5/thumbnails/2.jpg)
Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002
U.S.S. AlabamaMobile, Alabama
![Page 3: 2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.](https://reader035.fdocuments.us/reader035/viewer/2022070404/56649f335503460f94c50131/html5/thumbnails/3.jpg)
HWQ
Let f(x) and g(x) be 2 differentiable functions such that:
x F(x) G(x) F’(x) G’(x)
4 1 7 8 -8
3 -5 -3 -4 6
-5 2 -10 9 -1
Find the derivative of f(x)g(x) at x = -5. -92
![Page 4: 2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.](https://reader035.fdocuments.us/reader035/viewer/2022070404/56649f335503460f94c50131/html5/thumbnails/4.jpg)
Calculus Warm-Up
3cosd
xdx
23cos sinx x
![Page 5: 2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.](https://reader035.fdocuments.us/reader035/viewer/2022070404/56649f335503460f94c50131/html5/thumbnails/5.jpg)
Calculus Warm-Up
2 1d
xdx
We will come back to this problem later.
![Page 6: 2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.](https://reader035.fdocuments.us/reader035/viewer/2022070404/56649f335503460f94c50131/html5/thumbnails/6.jpg)
The Chain Rule
Copyright © Cengage Learning. All rights reserved.
2.4
![Page 7: 2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.](https://reader035.fdocuments.us/reader035/viewer/2022070404/56649f335503460f94c50131/html5/thumbnails/7.jpg)
Find the derivative of a composite function using the Chain Rule.
Objective:
![Page 8: 2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.](https://reader035.fdocuments.us/reader035/viewer/2022070404/56649f335503460f94c50131/html5/thumbnails/8.jpg)
We now have a pretty good list of “shortcuts” to find derivatives of simple functions.
Of course, many of the functions that we will encounter are not so simple. What is needed is a way to combine derivative rules to evaluate more complicated functions.
We do this with the chain rule.
![Page 9: 2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.](https://reader035.fdocuments.us/reader035/viewer/2022070404/56649f335503460f94c50131/html5/thumbnails/9.jpg)
Consider a simple composite function:
6 10y x
2 3 5y x
If 3 5u x
then 2y u
6 10y x 2y u 3 5u x
6dy
dx 2
dy
du 3
du
dx
dy dy du
dx du dx
6 2 3
![Page 10: 2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.](https://reader035.fdocuments.us/reader035/viewer/2022070404/56649f335503460f94c50131/html5/thumbnails/10.jpg)
and another:
5 2y u
where 3u t
then 5 3 2y t
3u t
15dy
dt 5
dy
du 3
du
dt
dy dy du
dt du dt
15 5 3
5 3 2y t
15 2y t
5 2y u
![Page 11: 2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.](https://reader035.fdocuments.us/reader035/viewer/2022070404/56649f335503460f94c50131/html5/thumbnails/11.jpg)
and another:29 6 1y x x
23 1y x
If 3 1u x
3 1u x
18 6dy
xdx
2dy
udu
3du
dx
dy dy du
dx du dx
2y u
2then y u
29 6 1y x x
2 3 1dy
xdu
6 2dy
xdu
18 6 6 2 3x x This pattern is called the chain rule.
![Page 12: 2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.](https://reader035.fdocuments.us/reader035/viewer/2022070404/56649f335503460f94c50131/html5/thumbnails/12.jpg)
![Page 13: 2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.](https://reader035.fdocuments.us/reader035/viewer/2022070404/56649f335503460f94c50131/html5/thumbnails/13.jpg)
dy dy du
dx du dx Chain Rule:
Example: sinf x x 2 4g x x Find: at 2f g x
( ( )) '( ( )) '( )df g x f g x g x
dx
2sin 4f g x x
or:
![Page 14: 2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.](https://reader035.fdocuments.us/reader035/viewer/2022070404/56649f335503460f94c50131/html5/thumbnails/14.jpg)
2sin 4y f g x x
2 2cos 4 4dy d
x xdx dx
2cos 4 2dy
x xdx
Differentiate the outside function...
…then the inside function
at 2, 4x y
( ( )) '( ( )) '( )df g x f g x g x
dxChain Rule:
2cos 2 4 2 2dy
dx
cos 0 4dy
dx
4dy
dx
22 cos 4dy
x xdx
![Page 15: 2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.](https://reader035.fdocuments.us/reader035/viewer/2022070404/56649f335503460f94c50131/html5/thumbnails/15.jpg)
Use the chain rule to differentiate:
3cosd
xdx
23cos sinx x
![Page 16: 2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.](https://reader035.fdocuments.us/reader035/viewer/2022070404/56649f335503460f94c50131/html5/thumbnails/16.jpg)
Use the chain rule to differentiate:
2 1d
xdx
2 1
x
x
![Page 17: 2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.](https://reader035.fdocuments.us/reader035/viewer/2022070404/56649f335503460f94c50131/html5/thumbnails/17.jpg)
Another example:
2cos 3d
xdx
2cos 3
dx
dx
2 cos 3 cos 3d
x xdx
derivative of theoutside function
derivative of theinside function
It looks like we need to use the chain rule again!
![Page 18: 2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.](https://reader035.fdocuments.us/reader035/viewer/2022070404/56649f335503460f94c50131/html5/thumbnails/18.jpg)
con’t:
2cos 3d
xdx
2cos 3
dx
dx
2 cos 3 cos 3d
x xdx
2cos 3 sin 3 3d
x x xdx
2cos 3 sin 3 3x x
6cos 3 sin 3x x
The chain rule can be used more than once.
(That’s what makes the “chain” in the “chain rule”!)
![Page 19: 2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.](https://reader035.fdocuments.us/reader035/viewer/2022070404/56649f335503460f94c50131/html5/thumbnails/19.jpg)
The most common mistake on derivative tests is to forget to use the chain rule.
Every derivative problem could be thought of as a chain-rule problem:
2dx
dx2d
x xdx
2 1x 2x
derivative of outside function
derivative of inside function
The derivative of x is one.
![Page 20: 2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.](https://reader035.fdocuments.us/reader035/viewer/2022070404/56649f335503460f94c50131/html5/thumbnails/20.jpg)
Practice:
Differentiate: 32 2f x x
226 2f x x x
![Page 21: 2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.](https://reader035.fdocuments.us/reader035/viewer/2022070404/56649f335503460f94c50131/html5/thumbnails/21.jpg)
Practice:
Differentiate: 2
7
2 3g t
t
3
28
2 3g t
t
![Page 22: 2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.](https://reader035.fdocuments.us/reader035/viewer/2022070404/56649f335503460f94c50131/html5/thumbnails/22.jpg)
sin 2 ' ?
' cos 2 2
' 2cos 2y x
y x y
y x
2
2
tan 3 ' ?
' sec 3 3
' 3sec 3
y x y
y x
y x
' sin 1
cos 1 ' ?
' sin 1 1
y x y
y x
y x
2cos 3 ' ?
' 2cos 3
' 2 cos 3
y x y
y x
y x
2
2
2
cos 3 ' ?
' sin 3
' 6 sin 3
6
y x y
y x x
y x x
2
2
2
cos 3 ' ?
' sin 3 2 3
' 18 si
3
n 9
y x y
x
x x
x
y
y
![Page 23: 2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.](https://reader035.fdocuments.us/reader035/viewer/2022070404/56649f335503460f94c50131/html5/thumbnails/23.jpg)
BC Homework
• 2.4 Day 1 p. 137: 7-31 odd, 41-57 odd,
67-71 odd, 81,83
• 2.4 Day 2: MMM pgs. 44-46
• 2.4 Day 3: MMM pg. 50
![Page 24: 2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.](https://reader035.fdocuments.us/reader035/viewer/2022070404/56649f335503460f94c50131/html5/thumbnails/24.jpg)
2.4 The Chain Rule – Day 2
Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002
![Page 25: 2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.](https://reader035.fdocuments.us/reader035/viewer/2022070404/56649f335503460f94c50131/html5/thumbnails/25.jpg)
HWQ
Differentiate:
3sin 4f t t
212sin 4 cos 4f t t t
![Page 26: 2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.](https://reader035.fdocuments.us/reader035/viewer/2022070404/56649f335503460f94c50131/html5/thumbnails/26.jpg)
2.4 Warm-up
223
1/
2
322 / 32
3 2
1 Where does ' 0?
Where does ' not exist?
2' 1 2
34
' 0@ 0
' when 1
1
3 1
0 1
f x x f x
f x
f x x x
x
x
f
f x x
x x
f x DNE x x
![Page 27: 2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.](https://reader035.fdocuments.us/reader035/viewer/2022070404/56649f335503460f94c50131/html5/thumbnails/27.jpg)
223 1
f x x
![Page 28: 2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.](https://reader035.fdocuments.us/reader035/viewer/2022070404/56649f335503460f94c50131/html5/thumbnails/28.jpg)
2 2
1/ 2 1/ 22 2 2
1/ 2 1/ 22 3 2
1/ 2 1/ 22 2 3
1/ 2 1/ 22 2
2 3
2
2
3
1/ 2
2
2
1 ' ?
1' 2 1 1 2
2
2
2 3
1 1
2 1 1
1 1
2 1
1
1
1
f x x x f x
f x x x x x x
x x x x
x x x x
x x
x x xx
f x x
x
x
x
x
Common Denominator
![Page 29: 2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.](https://reader035.fdocuments.us/reader035/viewer/2022070404/56649f335503460f94c50131/html5/thumbnails/29.jpg)
3 2
1/ 3 2 / 32 2
2 / 32
21/ 32
2 / 32
2 / 32
1/ 3 2 / 32 2 2
2 / 3 2 / 32 2
2 / 32
2 2
2
1/ 32
2
4
/ 32 2 2
2 / 3 4 /32 32/2
' ?4
11 4 4 2
3'4
24
3 4
4
3 4 4 2
3 4 3 4
4
3 4 2
3 4 3 4 12
3
2
4 3 4 4
4
xf x f x
x
x x x xf x
x
xx
x
x
x x x
x x
x
x x
x x x
x x
x
x
xf
x
x
![Page 30: 2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.](https://reader035.fdocuments.us/reader035/viewer/2022070404/56649f335503460f94c50131/html5/thumbnails/30.jpg)
HWQ (no calculator)
Determine the point(s) at which the graph of
has a horizontal tangent. 2 1
xf x
x
1,1
![Page 31: 2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.](https://reader035.fdocuments.us/reader035/viewer/2022070404/56649f335503460f94c50131/html5/thumbnails/31.jpg)
2
2
2
2
2 22
2 2
2 22 32
3 1 ' ?
3
3 3 3 1 23 1' 2
3 3
3 1 3 2 3 19 6 22
3
3 2
33
9
xy y
x
x x xxy
x x
x x x x x
x
x x
xx
![Page 32: 2.4 The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002.](https://reader035.fdocuments.us/reader035/viewer/2022070404/56649f335503460f94c50131/html5/thumbnails/32.jpg)
AB Homework
• 2.4 Day 1 p. 137: 1-31 odd, 41-57 odd
• 2.4 Day 2: p. 137: 59-73 odd, 79-89 odd
• 2.4 Day 3: MMM pgs. 44-46
• 2.4 Day 4: Chain Rule W/S