1. Calculus Rules

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

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

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

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

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

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

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

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

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

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

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

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

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

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

16.  v  y  x2  3

17.  v  y  x2  3
1
  x  3
2 2

18.  v  y  x2  3
1
  x  3
2 2
1
  x  3 2  2 x 
dy 1 2 
dx 2

19.  v  y  x2  3
1
  x  3
2 2
1
  x  3 2  2 x 
dy 1 2 
dx 2
1
 x  x  3
2 
2

20.  v  y  x2  3
1
  x  3
2 2
1
  x  3 2  2 x 
dy 1 2 
dx 2
1
 x  x  3
2 
2
x

x2  3

21.  v  y  x2  3 dy dy dt
 
1
  x  3
2 2 dx dt dx
1
  x  3 2  2 x 
dy 1 2 
dx 2
1
 x  x  3
2 
2
x

x2  3

22.  v  y  x2  3 dy dy dt
 
1
  x  3
2 2 dx dt dx
1
  x  3 2  2 x 
dy 1 2   vi  x  4t y  2t 2
dx 2
1
 x  x  3
2 
2
x

x2  3

23.  v  y  x2  3 dy dy dt
 
1
  x  3
2 2 dx dt dx
1
  x  3 2  2 x 
dy 1 2   vi  x  4t y  2t 2
dx 2 dx
1 4
 x  x  3
2  dt
2
x

x2  3

24.  v  y  x2  3 dy dy dt
 
1
  x  3
2 2 dx dt dx
1
  x  3 2  2 x 
dy 1 2   vi  x  4t y  2t 2
dx dy
dx 2
1 4  4t
 x  x  3
2  dt dt
2
x

x2  3

25.  v  y  x2  3 dy dy dt
 
1
  x  3
2 2 dx dt dx
1
  x  3 2  2 x 
dy 1 2   vi  x  4t y  2t 2
dx dy
dx 2
1 4  4t
 x  x  3
2  dt dt
2
dy dy dt
x  
 dx dt dx
x2  3

26.  v  y  x2  3 dy dy dt
 
1
  x  3
2 2 dx dt dx
1
  x  3 2  2 x 
dy 1 2   vi  x  4t y  2t 2
dx dy
dx 2
1 4  4t
 x  x  3
2  dt dt
2
dy dy dt
x  
 dx dt dx
x2  3 1
 4t 
4

27.  v  y  x2  3 dy dy dt
 
1
  x  3
2 2 dx dt dx
1
  x  3 2  2 x 
dy 1 2   vi  x  4t y  2t 2
dx dy
dx 2
1 4  4t
 x  x  3
2  dt dt
2
dy dy dt
x  
 dx dt dx
x2  3 1
 4t 
4
t

28.  v  y  x2  3 dy dy dt
 
1
  x  3
2 2 dx dt dx
1
  x  3 2  2 x 
dy 1 2   vi  x  4t y  2t 2
dx dy
dx 2
1 4  4t
 x  x  3
2  dt dt
2
dy dy dt
x  
 dx dt dx
x2  3 1
 4t 
4
t
Exercise 7E; 1adgj, 2behk, 3bd, 4adgj, 5ad, 6b, 7b, 8b,
10bdg, 11a, 13