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Calculus Rules
Calculus Rules
3. Quotient Rule
2
d u vu uv
dx v v
    
 
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
    
 
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
    
 
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
    
 
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
    
 
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


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


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


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

  


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

  


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


 
4. Reciprocal Rule 2
d k kv
dx v v
   
 
4. Reciprocal Rule 2
d k kv
dx v v
   
 
“MINUS the DERIVATIVE on the FUNCTION SQUARED”
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. 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


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


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


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



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



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

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11 x1 t09 06 quotient & reciprocal rules (2013)

  • 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