11X1 T08 04 chain rule

Calculus Rules

Calculus Rules1. Chain Rule 1 n nd

u nu udx

Calculus Rules1. Chain Rule 1 n nd

u nu udx

“BRING DOWN the POWER, LOWER the POWER, DIFF the INSIDE”

Calculus Rules1. Chain Rule 1 n nd

u nu udx

“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 udx

“BRING DOWN the POWER, LOWER the POWER, DIFF the INSIDE”

2e.g. 2 1i y x

12 2 1 2

dyx

dx

Calculus Rules1. Chain Rule 1 n nd

u nu udx

“BRING DOWN the POWER, LOWER the POWER, DIFF the INSIDE”

2e.g. 2 1i y x

12 2 1 2

dyx

dx

4 2 1x

Calculus Rules1. Chain Rule 1 n nd

u nu udx

“BRING DOWN the POWER, LOWER the POWER, DIFF the INSIDE”

2e.g. 2 1i y x

12 2 1 2

dyx

dx

4 2 1x

7 3 4ii y x

Calculus Rules1. Chain Rule 1 n nd

u nu udx

“BRING DOWN the POWER, LOWER the POWER, DIFF the INSIDE”

2e.g. 2 1i y x

12 2 1 2

dyx

dx

4 2 1x

7 3 4ii y x

67 3 4 3

dyx

dx

Calculus Rules1. Chain Rule 1 n nd

u nu udx

“BRING DOWN the POWER, LOWER the POWER, DIFF the INSIDE”

2e.g. 2 1i y x

12 2 1 2

dyx

dx

4 2 1x

7 3 4ii y x

67 3 4 3

dyx

dx

621 3 4x

Calculus Rules1. Chain Rule 1 n nd

u nu udx

“BRING DOWN the POWER, LOWER the POWER, DIFF the INSIDE”

2e.g. 2 1i y x

12 2 1 2

dyx

dx

4 2 1x

7 3 4ii y x

67 3 4 3

dyx

dx

621 3 4x

32 10iii y x

Calculus Rules1. Chain Rule 1 n nd

u nu udx

“BRING DOWN the POWER, LOWER the POWER, DIFF the INSIDE”

2e.g. 2 1i y x

12 2 1 2

dyx

dx

4 2 1x

7 3 4ii y x

67 3 4 3

dyx

dx

621 3 4x

32 10iii y x

223 10 2dy

x xdx

Calculus Rules1. Chain Rule 1 n nd

u nu udx

“BRING DOWN the POWER, LOWER the POWER, DIFF the INSIDE”

2e.g. 2 1i y x

12 2 1 2

dyx

dx

4 2 1x

7 3 4ii y x

67 3 4 3

dyx

dx

621 3 4x

32 10iii y x

223 10 2dy

x xdx

226 10x x

Calculus Rules1. Chain Rule 1 n nd

u nu udx

“BRING DOWN the POWER, LOWER the POWER, DIFF the INSIDE”

2e.g. 2 1i y x

12 2 1 2

dyx

dx

4 2 1x

7 3 4ii y x

67 3 4 3

dyx

dx

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 udx

“BRING DOWN the POWER, LOWER the POWER, DIFF the INSIDE”

2e.g. 2 1i y x

12 2 1 2

dyx

dx

4 2 1x

7 3 4ii y x

67 3 4 3

dyx

dx

621 3 4x

32 10iii y x

223 10 2dy

x xdx

226 10x x

6 5 4 2iv y x

530 4 2 2

dyx

dx

Calculus Rules1. Chain Rule 1 n nd

u nu udx

“BRING DOWN the POWER, LOWER the POWER, DIFF the INSIDE”

2e.g. 2 1i y x

12 2 1 2

dyx

dx

4 2 1x

7 3 4ii y x

67 3 4 3

dyx

dx

621 3 4x

32 10iii y x

223 10 2dy

x xdx

226 10x x

6 5 4 2iv y x

530 4 2 2

dyx

dx

560 4 2x

2 3v y x

1

2 23x

2 3v y x

1

2 23x

1

2 21

3 22

dyx x

dx

2 3v y x

1

2 23x

1

2 21

3 22

dyx x

dx

1

2 23x x

2 3v y x

1

2 23x

1

2 21

3 22

dyx x

dx

1

2 23x x

2 3v y x

2 3

x

x

1

2 23x

1

2 21

3 22

dyx x

dx

1

2 23x x

2 3v y x

2 3

x

x

dy dy dtdx dt dx

1

2 23x

1

2 21

3 22

dyx x

dx

1

2 23x x

2 3v y x

2 3

x

x

dy dy dtdx dt dx

2 4 2vi x t y t

1

2 23x

1

2 21

3 22

dyx x

dx

1

2 23x x

2 3v y x

2 3

x

x

dy dy dtdx dt dx

2 4 2vi x t y t

4dxdt

1

2 23x

1

2 21

3 22

dyx x

dx

1

2 23x x

2 3v y x

2 3

x

x

dy dy dtdx dt dx

2 4 2vi x t y t

4dxdt

4dy

tdt

1

2 23x

1

2 21

3 22

dyx x

dx

1

2 23x x

2 3v y x

2 3

x

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

dyx x

dx

1

2 23x x

2 3v y x

2 3

x

x

dy dy dtdx dt dx

2 4 2vi x t y t

4dxdt

4dy

tdt

dy dy dtdx dt dx

14

4t

1

2 23x

1

2 21

3 22

dyx x

dx

1

2 23x x

2 3v y x

2 3

x

x

dy dy dtdx dt dx

2 4 2vi x t y t

4dxdt

4dy

tdt

dy dy dtdx dt dx

14

4t

t

1

2 23x

1

2 21

3 22

dyx x

dx

1

2 23x x

2 3v y x

2 3

x

x

dy dy dtdx dt dx

2 4 2vi x t y t

4dxdt

4dy

tdt

dy dy dtdx dt dx

14

4t

t

Exercise 7E; 1adgj, 2behk, 3bd, 4adgj, 5ad, 6b, 7b, 8b,10bdg, 11a, 13