11x1 t09 04 chain rule (2011)
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Calculus Rules
Calculus Rules1. Chain Rule 1 n nd u nu u
dx
Calculus Rules1. Chain Rule 1 n nd u nu u
dx
“BRING DOWN the POWER, LOWER the POWER, DIFF the INSIDE”
Calculus Rules1. Chain Rule 1 n nd u nu u
dx
“BRING DOWN the POWER, LOWER the POWER, DIFF the INSIDE”
2e.g. 2 1i y x
Calculus Rules1. Chain Rule 1 n nd u nu u
dx
“BRING DOWN the POWER, LOWER the POWER, DIFF the INSIDE”
2e.g. 2 1i y x
12 2 1 2dy xdx
Calculus Rules1. Chain Rule 1 n nd u nu u
dx
“BRING DOWN the POWER, LOWER the POWER, DIFF the INSIDE”
2e.g. 2 1i y x
12 2 1 2dy xdx
4 2 1x
Calculus Rules1. Chain Rule 1 n nd u nu u
dx
“BRING DOWN the POWER, LOWER the POWER, DIFF the INSIDE”
2e.g. 2 1i y x
12 2 1 2dy xdx
4 2 1x
7 3 4ii y x
Calculus Rules1. Chain Rule 1 n nd u nu u
dx
“BRING DOWN the POWER, LOWER the POWER, DIFF the INSIDE”
2e.g. 2 1i y x
12 2 1 2dy xdx
4 2 1x
7 3 4ii y x
67 3 4 3dy xdx
Calculus Rules1. Chain Rule 1 n nd u nu u
dx
“BRING DOWN the POWER, LOWER the POWER, DIFF the INSIDE”
2e.g. 2 1i y x
12 2 1 2dy xdx
4 2 1x
7 3 4ii y x
67 3 4 3dy xdx
621 3 4x
Calculus Rules1. Chain Rule 1 n nd u nu u
dx
“BRING DOWN the POWER, LOWER the POWER, DIFF the INSIDE”
2e.g. 2 1i y x
12 2 1 2dy xdx
4 2 1x
7 3 4ii y x
67 3 4 3dy xdx
621 3 4x
32 10iii y x
Calculus Rules1. Chain Rule 1 n nd u nu u
dx
“BRING DOWN the POWER, LOWER the POWER, DIFF the INSIDE”
2e.g. 2 1i y x
12 2 1 2dy xdx
4 2 1x
7 3 4ii y x
67 3 4 3dy xdx
621 3 4x
32 10iii y x
223 10 2dy x xdx
Calculus Rules1. Chain Rule 1 n nd u nu u
dx
“BRING DOWN the POWER, LOWER the POWER, DIFF the INSIDE”
2e.g. 2 1i y x
12 2 1 2dy xdx
4 2 1x
7 3 4ii y x
67 3 4 3dy xdx
621 3 4x
32 10iii y x
223 10 2dy x xdx
226 10x x
Calculus Rules1. Chain Rule 1 n nd u nu u
dx
“BRING DOWN the POWER, LOWER the POWER, DIFF the INSIDE”
2e.g. 2 1i y x
12 2 1 2dy xdx
4 2 1x
7 3 4ii y x
67 3 4 3dy xdx
621 3 4x
32 10iii y x
223 10 2dy x xdx
226 10x x
6 5 4 2iv y x
Calculus Rules1. Chain Rule 1 n nd u nu u
dx
“BRING DOWN the POWER, LOWER the POWER, DIFF the INSIDE”
2e.g. 2 1i y x
12 2 1 2dy xdx
4 2 1x
7 3 4ii y x
67 3 4 3dy xdx
621 3 4x
32 10iii y x
223 10 2dy x xdx
226 10x x
6 5 4 2iv y x
530 4 2 2dy xdx
Calculus Rules1. Chain Rule 1 n nd u nu u
dx
“BRING DOWN the POWER, LOWER the POWER, DIFF the INSIDE”
2e.g. 2 1i y x
12 2 1 2dy xdx
4 2 1x
7 3 4ii y x
67 3 4 3dy xdx
621 3 4x
32 10iii y x
223 10 2dy x xdx
226 10x x
6 5 4 2iv y x
530 4 2 2dy xdx
560 4 2x
2 3v y x
1
2 23x
2 3v y x
1
2 23x
1
2 21 3 22
dy x xdx
2 3v y x
1
2 23x
1
2 21 3 22
dy x xdx
1
2 23x x
2 3v y x
1
2 23x
1
2 21 3 22
dy x xdx
1
2 23x x
2 3v y x
2 3x
x
1
2 23x
1
2 21 3 22
dy x xdx
1
2 23x x
2 3v y x
2 3x
x
dy dy dtdx dt dx
1
2 23x
1
2 21 3 22
dy x xdx
1
2 23x x
2 3v y x
2 3x
x
dy dy dtdx dt dx
2 4 2vi x t y t
1
2 23x
1
2 21 3 22
dy x xdx
1
2 23x x
2 3v y x
2 3x
x
dy dy dtdx dt dx
2 4 2vi x t y t
4dxdt
1
2 23x
1
2 21 3 22
dy x xdx
1
2 23x x
2 3v y x
2 3x
x
dy dy dtdx dt dx
2 4 2vi x t y t
4dxdt
4dy tdt
1
2 23x
1
2 21 3 22
dy x xdx
1
2 23x x
2 3v y x
2 3x
x
dy dy dtdx dt dx
2 4 2vi x t y t
4dxdt
4dy tdt
dy dy dtdx dt dx
1
2 23x
1
2 21 3 22
dy x xdx
1
2 23x x
2 3v y x
2 3x
x
dy dy dtdx dt dx
2 4 2vi x t y t
4dxdt
4dy tdt
dy dy dtdx dt dx
144
t
1
2 23x
1
2 21 3 22
dy x xdx
1
2 23x x
2 3v y x
2 3x
x
dy dy dtdx dt dx
2 4 2vi x t y t
4dxdt
4dy tdt
dy dy dtdx dt dx
144
t
t
1
2 23x
1
2 21 3 22
dy x xdx
1
2 23x x
2 3v y x
2 3x
x
dy dy dtdx dt dx
2 4 2vi x t y t
4dxdt
4dy tdt
dy dy dtdx dt dx
144
t
t
Exercise 7E; 1adgj, 2behk, 3bd, 4adgj, 5ad, 6b, 7b, 8b,10bdg, 11a, 13
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