The document discusses calculus quotient and reciprocal rules. It provides the quotient rule formula and explains that to use it you should "square the bottom, write down the bottom and differentiate the top, minus write down the top and differentiate the bottom." Examples are provided to demonstrate applying the quotient rule. The reciprocal rule formula is also given, along with the explanation that you should "minus the derivative of the function squared." More examples demonstrate applying the reciprocal rule.

1. Calculus Rules

2. Calculus Rules
3. Quotient Rule
2
d u vu uv
dx v v
    
 

3. Calculus Rules
3. Quotient Rule
“SQUARE the BOTTOM, write down the BOTTOM and DIFF the
TOP, MINUS write down the TOP and DIFF the BOTTOM”
2
d u vu uv
dx v v
    
 

4. Calculus Rules
3. Quotient Rule
“SQUARE the BOTTOM, write down the BOTTOM and DIFF the
TOP, MINUS write down the TOP and DIFF the BOTTOM”
 e.g.
1 2
x
i y
x


2
d u vu uv
dx v v
    
 

5. Calculus Rules
3. Quotient Rule
“SQUARE the BOTTOM, write down the BOTTOM and DIFF the
TOP, MINUS write down the TOP and DIFF the BOTTOM”
 e.g.
1 2
x
i y
x


     
 
2
1 2 1 2
1 2
x xdy
dx x
 


2
d u vu uv
dx v v
    
 

6. Calculus Rules
3. Quotient Rule
“SQUARE the BOTTOM, write down the BOTTOM and DIFF the
TOP, MINUS write down the TOP and DIFF the BOTTOM”
 e.g.
1 2
x
i y
x


     
 
2
1 2 1 2
1 2
x xdy
dx x
 


 
2
1 2 2
1 2
x x
x
 


2
d u vu uv
dx v v
    
 

7. Calculus Rules
3. Quotient Rule
“SQUARE the BOTTOM, write down the BOTTOM and DIFF the
TOP, MINUS write down the TOP and DIFF the BOTTOM”
 e.g.
1 2
x
i y
x


     
 
2
1 2 1 2
1 2
x xdy
dx x
 


 
2
1 2 2
1 2
x x
x
 


2
d u vu uv
dx v v
    
 
 
2
1
1 2x



8. Calculus Rules
3. Quotient Rule
“SQUARE the BOTTOM, write down the BOTTOM and DIFF the
TOP, MINUS write down the TOP and DIFF the BOTTOM”
 e.g.
1 2
x
i y
x


     
 
2
1 2 1 2
1 2
x xdy
dx x
 


 
2
1 2 2
1 2
x x
x
 


  2
2
4
x
ii y
x


2
d u vu uv
dx v v
    
 
 
2
1
1 2x



9. Calculus Rules
3. Quotient Rule
“SQUARE the BOTTOM, write down the BOTTOM and DIFF the
TOP, MINUS write down the TOP and DIFF the BOTTOM”
 e.g.
1 2
x
i y
x


     
 
2
1 2 1 2
1 2
x xdy
dx x
 


 
2
1 2 2
1 2
x x
x
 


  2
2
4
x
ii y
x


         
 
1 1
2 22 2
2
1
4 2 2 4 2
2
4
x x x x
dy
dx x
 
   
 

2
d u vu uv
dx v v
    
 
 
2
1
1 2x



10. Calculus Rules
3. Quotient Rule
“SQUARE the BOTTOM, write down the BOTTOM and DIFF the
TOP, MINUS write down the TOP and DIFF the BOTTOM”
 e.g.
1 2
x
i y
x


     
 
2
1 2 1 2
1 2
x xdy
dx x
 


 
2
1 2 2
1 2
x x
x
 


  2
2
4
x
ii y
x


         
 
1 1
2 22 2
2
1
4 2 2 4 2
2
4
x x x x
dy
dx x
 
   
 

2
d u vu uv
dx v v
    
 
 
2
1
1 2x


   
 
1 1
2 2 22 2
2
2 4 2 4
4
x x x
x

  



11. Calculus Rules
3. Quotient Rule
“SQUARE the BOTTOM, write down the BOTTOM and DIFF the
TOP, MINUS write down the TOP and DIFF the BOTTOM”
 e.g.
1 2
x
i y
x


     
 
2
1 2 1 2
1 2
x xdy
dx x
 


 
2
1 2 2
1 2
x x
x
 


  2
2
4
x
ii y
x


         
 
1 1
2 22 2
2
1
4 2 2 4 2
2
4
x x x x
dy
dx x
 
   
 

2
d u vu uv
dx v v
    
 
 
2
1
1 2x


   
 
1 1
2 2 22 2
2
2 4 2 4
4
x x x
x

  


    
 
1
2 2 22
2
2 4 4
4
x x x
x

  



12. Calculus Rules
3. Quotient Rule
“SQUARE the BOTTOM, write down the BOTTOM and DIFF the
TOP, MINUS write down the TOP and DIFF the BOTTOM”
 e.g.
1 2
x
i y
x


     
 
2
1 2 1 2
1 2
x xdy
dx x
 


 
2
1 2 2
1 2
x x
x
 


  2
2
4
x
ii y
x


         
 
1 1
2 22 2
2
1
4 2 2 4 2
2
4
x x x x
dy
dx x
 
   
 

2
d u vu uv
dx v v
    
 
 
2
1
1 2x


   
 
1 1
2 2 22 2
2
2 4 2 4
4
x x x
x

  


    
 
1
2 2 22
2
2 4 4
4
x x x
x

  


 2 2
8
4 4x x


 

13. 4. Reciprocal Rule 2
d k kv
dx v v
   
 

14. 4. Reciprocal Rule 2
d k kv
dx v v
   
 
“MINUS the DERIVATIVE on the FUNCTION SQUARED”

15. 4. Reciprocal Rule 2
d k kv
dx v v
   
 
“MINUS the DERIVATIVE on the FUNCTION SQUARED”
  2
1
e.g. i y
x


16. 4. Reciprocal Rule 2
d k kv
dx v v
   
 
“MINUS the DERIVATIVE on the FUNCTION SQUARED”
  2
1
e.g. i y
x

4
2dy x
dx x



17. 4. Reciprocal Rule 2
d k kv
dx v v
   
 
“MINUS the DERIVATIVE on the FUNCTION SQUARED”
  2
1
e.g. i y
x

4
2dy x
dx x


3
2
x



18. 4. Reciprocal Rule 2
d k kv
dx v v
   
 
“MINUS the DERIVATIVE on the FUNCTION SQUARED”
  2
1
e.g. i y
x

4
2dy x
dx x


3
2
x


  2
6
4 3
ii y
x



19. 4. Reciprocal Rule 2
d k kv
dx v v
   
 
“MINUS the DERIVATIVE on the FUNCTION SQUARED”
  2
1
e.g. i y
x

4
2dy x
dx x


3
2
x


  2
6
4 3
ii y
x


 
 
22
6 8
4 3
xdy
dx x




20. 4. Reciprocal Rule 2
d k kv
dx v v
   
 
“MINUS the DERIVATIVE on the FUNCTION SQUARED”
  2
1
e.g. i y
x

4
2dy x
dx x


3
2
x


  2
6
4 3
ii y
x


 
 
22
6 8
4 3
xdy
dx x



 
22
48
4 3
x
x




21. 4. Reciprocal Rule 2
d k kv
dx v v
   
 
“MINUS the DERIVATIVE on the FUNCTION SQUARED”
  2
1
e.g. i y
x

4
2dy x
dx x


3
2
x


  2
6
4 3
ii y
x


 
 
22
6 8
4 3
xdy
dx x



 
22
48
4 3
x
x



Exercise 7G; 1aceg, 2, 4a, 6a, 8a