1. Calculus Rules

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

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

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

5. Calculus Rules
d
2. Product Rule  uv   uv  vu
dx
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
e.g.  i  y  x 7  x 9  6 
  x 7  9 x8    x 9  6  7 x 6 
dy
dx

6. Calculus Rules
d
2. Product Rule  uv   uv  vu
dx
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
e.g.  i  y  x 7  x 9  6 
  x 7  9 x8    x 9  6  7 x 6 
dy
dx
 9 x15  7 x15  42 x 6

7. Calculus Rules
d
2. Product Rule  uv   uv  vu
dx
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
e.g.  i  y  x 7  x 9  6 
  x 7  9 x8    x 9  6  7 x 6 
dy
dx
 9 x15  7 x15  42 x 6
 16 x15  42 x 6

8. Calculus Rules
d
2. Product Rule  uv   uv  vu
dx
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
e.g.  i  y  x 7  x 9  6   ii  y   x  2  2 x  3
  x 7  9 x8    x 9  6  7 x 6 
dy
dx
 9 x15  7 x15  42 x 6
 16 x15  42 x 6

9. Calculus Rules
d
2. Product Rule  uv   uv  vu
dx
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
e.g.  i  y  x 7  x 9  6   ii  y   x  2  2 x  3
dy
  x  9 x    x  6  7 x    x  2  2    2 x  31
dy 7 8 9 6
dx dx
 9 x15  7 x15  42 x 6
 16 x15  42 x 6

10. Calculus Rules
d
2. Product Rule  uv   uv  vu
dx
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
e.g.  i  y  x 7  x 9  6   ii  y   x  2  2 x  3
dy
  x  9 x    x  6  7 x    x  2  2    2 x  31
dy 7 8 9 6
dx dx
 9 x15  7 x15  42 x 6  2x  4  2x  3
 16 x15  42 x 6

11. Calculus Rules
d
2. Product Rule  uv   uv  vu
dx
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
e.g.  i  y  x 7  x 9  6   ii  y   x  2  2 x  3
dy
  x  9 x    x  6  7 x    x  2  2    2 x  31
dy 7 8 9 6
dx dx
 9 x15  7 x15  42 x 6  2x  4  2x  3
 16 x15  42 x 6  4x  7

12. Calculus Rules
d
2. Product Rule  uv   uv  vu
dx
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
e.g.  i  y  x 7  x 9  6   ii  y   x  2  2 x  3
dy
  x  9 x    x  6  7 x    x  2  2    2 x  31
dy 7 8 9 6
dx dx
 9 x15  7 x15  42 x 6  2x  4  2x  3
 16 x15  42 x 6  4x  7
 iii 
d
dx
 x 7  x 3  3 x 2  7 

13. Calculus Rules
d
2. Product Rule  uv   uv  vu
dx
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
e.g.  i  y  x 7  x 9  6   ii  y   x  2  2 x  3
dy
  x  9 x    x  6  7 x    x  2  2    2 x  31
dy 7 8 9 6
dx dx
 9 x15  7 x15  42 x 6  2x  4  2x  3
 16 x15  42 x 6  4x  7
 iii 
d
dx
 x 7  x 3  3 x 2  7 
  x 7  x 3   6 x    3 x 2  7  7 x 6  3 x 2 

14. Calculus Rules
d
2. Product Rule  uv   uv  vu
dx
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
e.g.  i  y  x 7  x 9  6   ii  y   x  2  2 x  3
dy
  x  9 x    x  6  7 x    x  2  2    2 x  31
dy 7 8 9 6
dx dx
 9 x15  7 x15  42 x 6  2x  4  2x  3
 16 x15  42 x 6  4x  7
 iii 
d
dx
 x 7  x 3  3 x 2  7 
  x 7  x 3   6 x    3 x 2  7  7 x 6  3 x 2 
 6 x8  6 x 4  21x8  9 x 4  49 x 6  21x 2

15. Calculus Rules
d
2. Product Rule  uv   uv  vu
dx
“Write down the FIRST and DIFF the SECOND, PLUS write down the
SECOND and DIFF the FIRST”
e.g.  i  y  x 7  x 9  6   ii  y   x  2  2 x  3
dy
  x  9 x    x  6  7 x    x  2  2    2 x  31
dy 7 8 9 6
dx dx
 9 x15  7 x15  42 x 6  2x  4  2x  3
 16 x15  42 x 6  4x  7
 iii 
d
dx
 x 7  x 3  3 x 2  7 
  x 7  x 3   6 x    3 x 2  7  7 x 6  3 x 2 
 6 x8  6 x 4  21x8  9 x 4  49 x 6  21x 2
 27 x8  49 x 6  15 x 4  21x 2

16.  iv  y  3x  x  4 
2 5

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

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

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

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

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

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

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

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

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

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

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

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

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