1. Power Rule If H(x) = (3x 7 If H(x) = (3x 7 -4x) 12 /12 3x 7 then H’(x) = (3x 7 - 4x) 11 (21x 6 - 4) When differentiating, the first answer is multiplied by the derivative, g’(x).

2. Power Rule If H(x) = (3x 7 If H(x) = (3x 7 -4x) 12 /12 3x 7 then H’(x) = (3x 7 -4x) 11 (21x 6 -4) 3x 7 H is an antiderivative of (3x 7 -4x) 11 (21x 6 -4)

3. Anti Power Rule Thus that term must be removed when antidifferentiating.

4. Anti Power Rule copy derivative is here Add one Copy

5. Anti Power Rule Copy g(x) If derivative is here Add one to exponent Divide by new exponent

6. The derivative of (2x-3). What is 2?

7. The derivative of (2x-3).

8. Anti Power Rule Copy g(x) If derivative is here Add one to exponent Divide by new exponent

9. = A. (7x+2) 13 /13 B. (7x+2) 13 /13 + C C. (7x+2) 11 /13 + C D. (7x+2) 11 /11 + C

10. = A. (7x+2) 13 /13 B. (7x+2) 13 /13 + C C. (7x+2) 11 /13 + C D. (7x+2) 11 /11 + C

11. What is the derivative of 12x + 3 ?

12. Number of cable telephones 3.2 million in 2004(t=0) Rate of growth r(t) = 3.36(t+1) 0.05 Rate of growth r(t) = 3.36(t+1) 0.05 How fast is it growing in 2008? How fast is it growing in 2008? How many cable phones in 2008? How many cable phones in 2008?

13. Rate r(t)=3.36 (t+1) 0.05. How fast is it growing in 2008? r(4) = 3.36(4+1) 0.05 = million/yr r(4) = 3.36(4+1) 0.05 = million/yr

14. r(t)=3.36 (t+1) 0.05 Find r(4) = mill./yr. 3.640.2

15. Number of cable telephones 3.2 million in 2004(t=0) Rate of growth r(t) = 3.36(t+1) 0.05 Rate of growth r(t) = 3.36(t+1) 0.05 How fast is it growing in 2008? How fast is it growing in 2008? How many cable phones in 2008? How many cable phones in 2008?

16. N(t) = A. N(t)=. B. N(t). C. N(t)

17. N(t) = A. N(t)=. B. N(t). C. N(t)

18. Rate r(t)=3.36 (t+1) 0.05. How many phones are there in 2008? N(t) = N(t) = = 3.36(t+1) 1.05 /1.05 + C = 3.36(t+1) 1.05 /1.05 + C = 3.2(t+1) 1.05 + C = 3.2(t+1) 1.05 + C N(0) = 3.2 + C = 3.2 so N(0) = 3.2 + C = 3.2 so

19. N(0) = 3.2 + C = 3.2 C = million

20. 0.00.1

21. N(t) = 3.2(t+1) 1.05. How many phones are there in 2008? N(t) = 3.2(t+1) 1.05 N(4) = 3.2(5) 1.05 N(4) = million subscribers

22. N(t) = 3.2(t+1) 1.05 N(4) = millions

23. 17.341.0

24. =

25. = -10.00.1

26. What is the derivative of 4x 2 + 12x + 3 ?

27. What is the exponent?

28. Now it is in the correct form.

29. Anti Power Rule Copy 4x 2 +12x+3 Derivative is here Add one to exponent Divide by new exponent

31. Anti Power Rule Copy 5x 3 - 1 Derivative is here Add one to exponent Divide by new exponent

32. Anti Power Rule If H(x) = (3x 7 If H(x) = (3x 7 -4x) 12 3x 7 H’(x) = 12(3x 7 -4x) 11 (21x 6 -4)

33. Chain Rule If H(x) =F(g(x)) then H’(x) = F’(g(x)) g’(x) H(x) is an antiderivative of F ’(g(x)) g’(x)

34. Anti – Chain Rule Make sure g’(x) is there Make sure g’(x) is there Antidifferentiate F’ giving F Antidifferentiate F’ giving F Copy g(x) giving F(g(x)) Copy g(x) giving F(g(x))

35. Anti Chain Rule

36. Antidifferentiate cos(x) Derivative of x 2 is here Copy x 2 Copy x 2

37. Anti Chain Rule Antidifferentiate sec(x)tan(x) sec(x)tan(x) Derivative of 9x is here Copy 9x Copy 9x

38. Anti Chain Rule Derivative of 3x is not here!

39. Anti Chain Rule Now it is in the correct form.

40. Anti Chain Rule Antidifferentiate cos(x) cos(x) Derivative of 3x is here! Copy 3x Copy 3x

41. Anti Chain Rule Antidifferentiate [sec(x)] 2 =sec 2 (x) [sec(x)] 2 =sec 2 (x) Derivative of sin(x) is here Copy sin(x) Copy sin(x)

42. = =

43. =

44. = 0.50.1