1. Calculus Rules

2. Calculus Rules
2. Product Rule  
d
uv uv vu
dx
  

3. Calculus Rules
2. Product Rule
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
 
d
uv uv vu
dx
  

4. Calculus Rules
2. Product Rule
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
   7 9
e.g. 6i y x x 
 
d
uv uv vu
dx
  

5. Calculus Rules
2. Product Rule
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
   7 9
e.g. 6i y x x 
     7 8 9 6
9 6 7
dy
x x x x
dx
  
 
d
uv uv vu
dx
  

6. Calculus Rules
2. Product Rule
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
   7 9
e.g. 6i y x x 
     7 8 9 6
9 6 7
dy
x x x x
dx
  
15 15 6
9 7 42x x x  
 
d
uv uv vu
dx
  

7. Calculus Rules
2. Product Rule
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
   7 9
e.g. 6i y x x 
     7 8 9 6
9 6 7
dy
x x x x
dx
  
15 15 6
9 7 42x x x  
 
d
uv uv vu
dx
  
15 6
16 42x x 

8. Calculus Rules
2. Product Rule
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
   7 9
e.g. 6i y x x 
     7 8 9 6
9 6 7
dy
x x x x
dx
  
15 15 6
9 7 42x x x  
    2 2 3ii y x x  
 
d
uv uv vu
dx
  
15 6
16 42x x 

9. Calculus Rules
2. Product Rule
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
   7 9
e.g. 6i y x x 
     7 8 9 6
9 6 7
dy
x x x x
dx
  
15 15 6
9 7 42x x x  
    2 2 3ii y x x  
     2 2 2 3 1
dy
x x
dx
   
 
d
uv uv vu
dx
  
15 6
16 42x x 

10. Calculus Rules
2. Product Rule
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
   7 9
e.g. 6i y x x 
     7 8 9 6
9 6 7
dy
x x x x
dx
  
15 15 6
9 7 42x x x  
    2 2 3ii y x x  
     2 2 2 3 1
dy
x x
dx
   
2 4 2 3x x   
 
d
uv uv vu
dx
  
15 6
16 42x x 

11. Calculus Rules
2. Product Rule
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
   7 9
e.g. 6i y x x 
     7 8 9 6
9 6 7
dy
x x x x
dx
  
15 15 6
9 7 42x x x  
    2 2 3ii y x x  
     2 2 2 3 1
dy
x x
dx
   
2 4 2 3x x   
 
d
uv uv vu
dx
  
15 6
16 42x x  4 7x 

12. Calculus Rules
2. Product Rule
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
   7 9
e.g. 6i y x x 
     7 8 9 6
9 6 7
dy
x x x x
dx
  
15 15 6
9 7 42x x x  
    2 2 3ii y x x  
     2 2 2 3 1
dy
x x
dx
   
2 4 2 3x x   
     7 3 2
3 7
d
iii x x x
dx
 
 
d
uv uv vu
dx
  
15 6
16 42x x  4 7x 

13. Calculus Rules
2. Product Rule
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
   7 9
e.g. 6i y x x 
     7 8 9 6
9 6 7
dy
x x x x
dx
  
15 15 6
9 7 42x x x  
    2 2 3ii y x x  
     2 2 2 3 1
dy
x x
dx
   
2 4 2 3x x   
     7 3 2
3 7
d
iii x x x
dx
 
     7 3 2 6 2
6 3 7 7 3x x x x x x    
 
d
uv uv vu
dx
  
15 6
16 42x x  4 7x 

14. Calculus Rules
2. Product Rule
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
   7 9
e.g. 6i y x x 
     7 8 9 6
9 6 7
dy
x x x x
dx
  
15 15 6
9 7 42x x x  
    2 2 3ii y x x  
     2 2 2 3 1
dy
x x
dx
   
2 4 2 3x x   
     7 3 2
3 7
d
iii x x x
dx
 
     7 3 2 6 2
6 3 7 7 3x x x x x x    
8 4 8 4 6 2
6 6 21 9 49 21x x x x x x     
 
d
uv uv vu
dx
  
15 6
16 42x x  4 7x 

15. Calculus Rules
2. Product Rule
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
   7 9
e.g. 6i y x x 
     7 8 9 6
9 6 7
dy
x x x x
dx
  
15 15 6
9 7 42x x x  
    2 2 3ii y x x  
     2 2 2 3 1
dy
x x
dx
   
2 4 2 3x x   
     7 3 2
3 7
d
iii x x x
dx
 
     7 3 2 6 2
6 3 7 7 3x x x x x x    
8 4 8 4 6 2
6 6 21 9 49 21x x x x x x     
 
d
uv uv vu
dx
  
15 6
16 42x x  4 7x 
8 6 4 2
27 49 15 21x x x x   

16.    
52
3 4iv y x x 

17.           
4 52 2
3 5 4 2 4 3
dy
x x x x
dx
   
   
52
3 4iv y x x 

18.           
4 52 2
3 5 4 2 4 3
dy
x x x x
dx
   
   
52
3 4iv y x x 
   
4 52 2 2
30 4 3 4x x x   

19.           
4 52 2
3 5 4 2 4 3
dy
x x x x
dx
   
   
52
3 4iv y x x 
   
4 52 2 2
30 4 3 4x x x   
    
42 2 2
4 30 3 4x x x   

20.           
4 52 2
3 5 4 2 4 3
dy
x x x x
dx
   
   
52
3 4iv y x x 
   
4 52 2 2
30 4 3 4x x x   
    
42 2 2
4 30 3 4x x x   
   
42 2
4 33 12x x  

21.           
4 52 2
3 5 4 2 4 3
dy
x x x x
dx
   
   
52
3 4iv y x x 
   
4 52 2 2
30 4 3 4x x x   
    
42 2 2
4 30 3 4x x x   
   
42 2
4 33 12x x  
   
42 2
3 4 11 4x x  

22.           
4 52 2
3 5 4 2 4 3
dy
x x x x
dx
   
   
52
3 4iv y x x 
  2 2 1v y x x 
   
4 52 2 2
30 4 3 4x x x   
    
42 2 2
4 30 3 4x x x   
   
42 2
4 33 12x x  
   
42 2
3 4 11 4x x  

23.           
4 52 2
3 5 4 2 4 3
dy
x x x x
dx
   
   
52
3 4iv y x x 
  2 2 1v y x x 
   
4 52 2 2
30 4 3 4x x x   
    
42 2 2
4 30 3 4x x x   
   
42 2
4 33 12x x  
   
42 2
3 4 11 4x x  
 
1
22 2 1x x 

24.           
4 52 2
3 5 4 2 4 3
dy
x x x x
dx
   
   
52
3 4iv y x x 
  2 2 1v y x x 
          
1 1
2 2
1
2 2 1 2 2 1 2
2
dy
x x x
dx

   
   
4 52 2 2
30 4 3 4x x x   
    
42 2 2
4 30 3 4x x x   
   
42 2
4 33 12x x  
   
42 2
3 4 11 4x x  
 
1
22 2 1x x 

25.           
4 52 2
3 5 4 2 4 3
dy
x x x x
dx
   
   
52
3 4iv y x x 
  2 2 1v y x x 
          
1 1
2 2
1
2 2 1 2 2 1 2
2
dy
x x x
dx

   
   
4 52 2 2
30 4 3 4x x x   
    
42 2 2
4 30 3 4x x x   
   
42 2
4 33 12x x  
   
42 2
3 4 11 4x x  
 
1
22 2 1x x 
    
1 1
2 22 2 1 2 2 1x x x

   

26.           
4 52 2
3 5 4 2 4 3
dy
x x x x
dx
   
   
52
3 4iv y x x 
  2 2 1v y x x 
          
1 1
2 2
1
2 2 1 2 2 1 2
2
dy
x x x
dx

   
   
4 52 2 2
30 4 3 4x x x   
    
42 2 2
4 30 3 4x x x   
   
42 2
4 33 12x x  
   
42 2
3 4 11 4x x  
 
1
22 2 1x x 
    
1 1
2 22 2 1 2 2 1x x x

   
    
1
22 2 1 2 1x x x

   

27.           
4 52 2
3 5 4 2 4 3
dy
x x x x
dx
   
   
52
3 4iv y x x 
  2 2 1v y x x 
          
1 1
2 2
1
2 2 1 2 2 1 2
2
dy
x x x
dx

   
   
4 52 2 2
30 4 3 4x x x   
    
42 2 2
4 30 3 4x x x   
   
42 2
4 33 12x x  
   
42 2
3 4 11 4x x  
 
1
22 2 1x x 
    
1 1
2 22 2 1 2 2 1x x x

   
    
1
22 2 1 2 1x x x

   
   
1
22 2 1 3 1x x

  

28.           
4 52 2
3 5 4 2 4 3
dy
x x x x
dx
   
   
52
3 4iv y x x 
  2 2 1v y x x 
          
1 1
2 2
1
2 2 1 2 2 1 2
2
dy
x x x
dx

   
   
4 52 2 2
30 4 3 4x x x   
    
42 2 2
4 30 3 4x x x   
   
42 2
4 33 12x x  
   
42 2
3 4 11 4x x  
 
1
22 2 1x x 
    
1 1
2 22 2 1 2 2 1x x x

   
    
1
22 2 1 2 1x x x

   
   
1
22 2 1 3 1x x

  
 2 3 1
2 1
xdy
dx x




29.           
4 52 2
3 5 4 2 4 3
dy
x x x x
dx
   
   
52
3 4iv y x x 
  2 2 1v y x x 
          
1 1
2 2
1
2 2 1 2 2 1 2
2
dy
x x x
dx

   
Exercise 7F; 1ac, 2bdf,
3a, 4ad, 5, 6ac,
7, 9, 13a*
   
4 52 2 2
30 4 3 4x x x   
    
42 2 2
4 30 3 4x x x   
   
42 2
4 33 12x x  
   
42 2
3 4 11 4x x  
 
1
22 2 1x x 
    
1 1
2 22 2 1 2 2 1x x x

   
    
1
22 2 1 2 1x x x

   
   
1
22 2 1 3 1x x

  
 2 3 1
2 1
xdy
dx x


