1. Calculus Rules

2. Calculus Rules
1. Chain Rule   1n nd
u nu u
dx

 

3. Calculus Rules
1. Chain Rule   1n nd
u nu u
dx

 
“BRING DOWN the POWER, LOWER the POWER, DIFF the
INSIDE”

4. Calculus Rules
1. Chain Rule   1n nd
u nu u
dx

 
“BRING DOWN the POWER, LOWER the POWER, DIFF the
INSIDE”
   
2
e.g. 2 1i y x 

5. Calculus Rules
1. Chain Rule   1n nd
u nu u
dx

 
“BRING DOWN the POWER, LOWER the POWER, DIFF the
INSIDE”
   
2
e.g. 2 1i y x 
   
1
2 2 1 2
dy
x
dx
 

6. Calculus Rules
1. Chain Rule   1n nd
u nu u
dx

 
“BRING DOWN the POWER, LOWER the POWER, DIFF the
INSIDE”
   
2
e.g. 2 1i y x 
   
1
2 2 1 2
dy
x
dx
 
 4 2 1x 

7. Calculus Rules
1. Chain Rule   1n nd
u nu u
dx

 
“BRING DOWN the POWER, LOWER the POWER, DIFF the
INSIDE”
   
2
e.g. 2 1i y x 
   
1
2 2 1 2
dy
x
dx
 
 4 2 1x 
   
7
3 4ii y x 

8. Calculus Rules
1. Chain Rule   1n nd
u nu u
dx

 
“BRING DOWN the POWER, LOWER the POWER, DIFF the
INSIDE”
   
2
e.g. 2 1i y x 
   
1
2 2 1 2
dy
x
dx
 
 4 2 1x 
   
7
3 4ii y x 
   
6
7 3 4 3
dy
x
dx
 

9. Calculus Rules
1. Chain Rule   1n nd
u nu u
dx

 
“BRING DOWN the POWER, LOWER the POWER, DIFF the
INSIDE”
   
2
e.g. 2 1i y x 
   
1
2 2 1 2
dy
x
dx
 
 4 2 1x 
   
7
3 4ii y x 
   
6
7 3 4 3
dy
x
dx
 
 
6
21 3 4x 

10. Calculus Rules
1. Chain Rule   1n nd
u nu u
dx

 
“BRING DOWN the POWER, LOWER the POWER, DIFF the
INSIDE”
   
2
e.g. 2 1i y x 
   
1
2 2 1 2
dy
x
dx
 
 4 2 1x 
   
7
3 4ii y x 
   
6
7 3 4 3
dy
x
dx
 
 
6
21 3 4x 
   
32
10iii y x 

11. Calculus Rules
1. Chain Rule   1n nd
u nu u
dx

 
“BRING DOWN the POWER, LOWER the POWER, DIFF the
INSIDE”
   
2
e.g. 2 1i y x 
   
1
2 2 1 2
dy
x
dx
 
 4 2 1x 
   
7
3 4ii y x 
   
6
7 3 4 3
dy
x
dx
 
 
6
21 3 4x 
   
32
10iii y x 
   
22
3 10 2
dy
x x
dx
 

12. Calculus Rules
1. Chain Rule   1n nd
u nu u
dx

 
“BRING DOWN the POWER, LOWER the POWER, DIFF the
INSIDE”
   
2
e.g. 2 1i y x 
   
1
2 2 1 2
dy
x
dx
 
 4 2 1x 
   
7
3 4ii y x 
   
6
7 3 4 3
dy
x
dx
 
 
6
21 3 4x 
   
32
10iii y x 
   
22
3 10 2
dy
x x
dx
 
 
22
6 10x x 

13. Calculus Rules
1. Chain Rule   1n nd
u nu u
dx

 
“BRING DOWN the POWER, LOWER the POWER, DIFF the
INSIDE”
   
2
e.g. 2 1i y x 
   
1
2 2 1 2
dy
x
dx
 
 4 2 1x 
   
7
3 4ii y x 
   
6
7 3 4 3
dy
x
dx
 
 
6
21 3 4x 
   
32
10iii y x 
   
22
3 10 2
dy
x x
dx
 
 
22
6 10x x 
   
6
5 4 2iv y x 

14. Calculus Rules
1. Chain Rule   1n nd
u nu u
dx

 
“BRING DOWN the POWER, LOWER the POWER, DIFF the
INSIDE”
   
2
e.g. 2 1i y x 
   
1
2 2 1 2
dy
x
dx
 
 4 2 1x 
   
7
3 4ii y x 
   
6
7 3 4 3
dy
x
dx
 
 
6
21 3 4x 
   
32
10iii y x 
   
22
3 10 2
dy
x x
dx
 
 
22
6 10x x 
   
6
5 4 2iv y x 
   
5
30 4 2 2
dy
x
dx
  

15. Calculus Rules
1. Chain Rule   1n nd
u nu u
dx

 
“BRING DOWN the POWER, LOWER the POWER, DIFF the
INSIDE”
   
2
e.g. 2 1i y x 
   
1
2 2 1 2
dy
x
dx
 
 4 2 1x 
   
7
3 4ii y x 
   
6
7 3 4 3
dy
x
dx
 
 
6
21 3 4x 
   
32
10iii y x 
   
22
3 10 2
dy
x x
dx
 
 
22
6 10x x 
   
6
5 4 2iv y x 
   
5
30 4 2 2
dy
x
dx
  
 
5
60 4 2x  

16.   2
3v y x 

17.  
1
2 23x 
  2
3v y x 

18.  
1
2 23x 
   
1
2 2
1
3 2
2
dy
x x
dx

 
  2
3v y x 

19.  
1
2 23x 
   
1
2 2
1
3 2
2
dy
x x
dx

 
 
1
2 23x x

 
  2
3v y x 

20.  
1
2 23x 
   
1
2 2
1
3 2
2
dy
x x
dx

 
 
1
2 23x x

 
  2
3v y x 
2
3
x
x



21.  
1
2 23x 
   
1
2 2
1
3 2
2
dy
x x
dx

 
 
1
2 23x x

 
  2
3v y x 
2
3
x
x


dy dy dt
dx dt dx
 

22.  
1
2 23x 
   
1
2 2
1
3 2
2
dy
x x
dx

 
 
1
2 23x x

 
  2
3v y x 
2
3
x
x


dy dy dt
dx dt dx
 
  2
4 2vi x t y t 

23.  
1
2 23x 
   
1
2 2
1
3 2
2
dy
x x
dx

 
 
1
2 23x x

 
  2
3v y x 
2
3
x
x


dy dy dt
dx dt dx
 
  2
4 2vi x t y t 
4
dx
dt


24.  
1
2 23x 
   
1
2 2
1
3 2
2
dy
x x
dx

 
 
1
2 23x x

 
  2
3v y x 
2
3
x
x


dy dy dt
dx dt dx
 
  2
4 2vi x t y t 
4
dx
dt
 4
dy
t
dt


25.  
1
2 23x 
   
1
2 2
1
3 2
2
dy
x x
dx

 
 
1
2 23x x

 
  2
3v y x 
2
3
x
x


dy dy dt
dx dt dx
 
  2
4 2vi x t y t 
4
dx
dt
 4
dy
t
dt

dy dy dt
dx dt dx
 

26.  
1
2 23x 
   
1
2 2
1
3 2
2
dy
x x
dx

 
 
1
2 23x x

 
  2
3v y x 
2
3
x
x


dy dy dt
dx dt dx
 
  2
4 2vi x t y t 
4
dx
dt
 4
dy
t
dt

dy dy dt
dx dt dx
 
1
4
4
t 

27.  
1
2 23x 
   
1
2 2
1
3 2
2
dy
x x
dx

 
 
1
2 23x x

 
  2
3v y x 
2
3
x
x


dy dy dt
dx dt dx
 
  2
4 2vi x t y t 
4
dx
dt
 4
dy
t
dt

dy dy dt
dx dt dx
 
1
4
4
t 
t

28.  
1
2 23x 
   
1
2 2
1
3 2
2
dy
x x
dx

 
 
1
2 23x x

 
  2
3v y x 
2
3
x
x


dy dy dt
dx dt dx
 
  2
4 2vi x t y t 
4
dx
dt
 4
dy
t
dt

dy dy dt
dx dt dx
 
1
4
4
t 
t
Exercise 7E; 1adgj, 2behk, 3bd, 4adgj, 5ad, 6b, 7b, 8b,
10bdg, 11a, 13