1. Derivatives of Exponential and Logarithmic Functions
Section 3.2
Derivatives of
Exponential and Logarithmic Functions

2. Exponential Growth
What are some things you can think of that grow exponentially?

3. Examples of Exponential Growth and Decay
Human Population growth (up to a point…)

4. Examples of Exponential Growth and Decay
Growth of Bacteria in a Petri Dish (again, up to a point)
Decay of radioactive atoms (Carbon-14)
Removal of drugs from body
Example: 50% of caffeine is removed from your body approximately every 4 hours

5. Sec 3.2 Derivatives of Exponential and Logarithmic Functions
Formulas for derivatives of:

6. Derivative of 𝑓 𝑥 = 𝑒 𝑥
Find f’(1) for various values of h

7. Derivative of 𝑓 𝑥 = 𝑒 𝑥 Find f’(1) for various values of h
𝑓 ′ 1 = lim ℎ→0 𝑓(𝑥+ℎ)−𝑓(𝑥) ℎ = lim ℎ→0 𝑓 −𝑓(1) 0.1

8. Derivative of 𝑓 𝑥 = 𝑒 𝑥 Find f’(1) for various values of h
𝑓 ′ 1 = lim ℎ→0 𝑓 −𝑓(1) =𝑙𝑖𝑚 ℎ→0 𝑒 − 𝑒 =2.859

9. Derivative of 𝑓 𝑥 = 𝑒 𝑥 Find f’(1) for various values of h
𝑓 ′ 1 =𝑙𝑖𝑚 ℎ→0 𝑒 − 𝑒 =2.859
𝑓 ′ 1 =𝑙𝑖𝑚 ℎ→0 𝑒 − 𝑒 =2.732
Smaller value of h = 0.01

10. Derivative of 𝑓 𝑥 = 𝑒 𝑥 Find f’(1) for various values of h
𝑓 ′ 1 =𝑙𝑖𝑚 ℎ→0 𝑒 − 𝑒 =2.859
𝑓 ′ 1 =𝑙𝑖𝑚 ℎ→0 𝑒 − 𝑒 =2.732
𝑓 ′ 1 =𝑙𝑖𝑚 ℎ→0 𝑒 − 𝑒 =2.720
Smaller value of h = 0.001

11. Derivative of 𝑓 𝑥 = 𝑒 𝑥 Find f’(1) for various values of h
𝑓 ′ 1 =𝑙𝑖𝑚 ℎ→0 𝑒 − 𝑒 =2.859
𝑓 ′ 1 =𝑙𝑖𝑚 ℎ→0 𝑒 − 𝑒 =2.732
𝑓 ′ 1 =𝑙𝑖𝑚 ℎ→0 𝑒 − 𝑒 =2.720
𝑓 ′ 1 =𝑙𝑖𝑚 ℎ→0 𝑒 − 𝑒 =2.718
Smaller value of h =

12. Derivative of 𝑓 𝑥 = 𝑒 𝑥 Find f’(1) for various values of h
The number e is a famous irrational number, and is one of the most important numbers in mathematics. The first few digits are:
(and more ...)
It is often called Euler's number after Leonhard Euler. 
e is the base of the Natural Logarithms (invented by John Napier).
Find f’(1) for various values of h
𝑓 ′ 1 =𝑙𝑖𝑚 ℎ→0 𝑒 − 𝑒 =2.859
𝑓 ′ 1 =𝑙𝑖𝑚 ℎ→0 𝑒 − 𝑒 =2.732
𝑓 ′ 1 =𝑙𝑖𝑚 ℎ→0 𝑒 − 𝑒 =2.720
𝑓 ′ 1 =𝑙𝑖𝑚 ℎ→0 𝑒 − 𝑒 =2.718
𝑓 ′ 1 =𝑙𝑖𝑚 ℎ→0 𝑒 − 𝑒 =2.718
Getting close to “e”

13. Derivative Function of 𝑓 𝑥 = 𝑒 𝑥
Start with the limit definition of the derivative:

14. Derivative Function of 𝑓 𝑥 = 𝑒 𝑥
Start with the limit definition of the derivative:

15. Derivative Function of 𝑓 𝑥 = 𝑒 𝑥
Start with the limit definition of the derivative:

16. Derivative Function of 𝑓 𝑥 = 𝑒 𝑥
Start with the limit definition of the derivative:

17. Derivative Function of 𝑓 𝑥 = 𝑒 𝑥
Start with the limit definition of the derivative:

18. Try various small values of h to find limiting value
-0.01
-0.001
0.001
0.01
0.995
0.9995 ?
1.0005
1.005
Try various small values of h to find limiting value

19. h
-0.01
-0.001
0.001
0.01
0.995
0.9995 1
1.0005
1.005
Limiting Value

20. h
-0.01
-0.001
0.001
0.01
0.995
0.9995 1
1.0005
1.005

21. So h
-0.01
-0.001
0.001
0.01
0.995
0.9995 ?
1.0005
1.005

22. Example: h
-0.01
-0.001
0.001
0.01
0.995
0.9995 ?
1.0005
1.005

23. So Example: h
-0.01
-0.001
0.001
0.01
0.995
0.9995 ?
1.0005
1.005

24. So Example: h
-0.01
-0.001
0.001
0.01
0.995
0.9995 ?
1.0005
1.005

25. Derivative Function for 𝑓 𝑥 = 𝑎 𝑥

26. Derivative Function for 𝑓 𝑥 = 𝑎 𝑥
Need the “chain rule” (coming up) to understand and prove (later)

27. Derivative Function for 𝑓 𝑥 = 𝑎 𝑥
Examples:

28. Derivative Function for 𝑓 𝑥 = 𝑎 𝑥
Examples:

29. Derivative Function for 𝑓 𝑥 = 𝑎 𝑥
Examples:

30. Derivative Function for 𝑓 𝑥 = 𝑎 𝑥
Examples:

31. Derivative Function for 𝑓 𝑡 = 𝑒 𝑘𝑡

32. Derivative Function for 𝑓 𝑡 = 𝑒 𝑘𝑡
Need the “chain rule” (coming up) to understand and prove (later)

33. Derivative Function for 𝑓 𝑡 = 𝑒 𝑘𝑡

34. Derivative Function for 𝑓 𝑥 =ln⁡(𝑥)

35. Derivative Function for 𝑓 𝑥 =ln⁡(𝑥)
Will also prove after chain rule

36. Derivative Function for 𝑓 𝑥 =ln⁡(𝑥)

37. Derivative Function for 𝑓 𝑥 =ln⁡(𝑥)
Will also prove after chain rule

38. Derivative Function for 𝑓 𝑥 =ln⁡(𝑥)
Will also prove after chain rule

39. Derivative Function for 𝑓 𝑥 =ln⁡(𝑥)
Will also prove after chain rule

40. Derivative Function for 𝑓 𝑥 =ln⁡(𝑥)
Will also prove after chain rule

41. Applications Example 1
Population of Nevada, P, in millions can be approximated by 𝑃= 𝑡
Where t is years since the start of 2000.

42. Applications Example 1
Population of Nevada, P, in millions can be approximated by 𝑃= 𝑡
Where t is years since the start of 2000.
At what rate was the population growing at the beginning of 2009?

43. Applications Example 1
Population of Nevada, P, in millions can be approximated by 𝑃= 𝑡
Where t is years since the start of 2000.
At what rate was the population growing at the beginning of 2009?
Instantaneous Rate of Change at t = 9

44. Applications Example 1
Population of Nevada, P, in millions can be approximated by 𝑃= 𝑡
Where t is years since the start of 2000.
At what rate was the population growing at the beginning of 2009?
Instantaneous Rate of Change at t = 9
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑡𝑖𝑚𝑒 = 𝑑𝑃 𝑑𝑡

45. Applications Example 1
Population of Nevada, P, in millions can be approximated by 𝑃= 𝑡
Where t is years since the start of 2000.
At what rate was the population growing at the beginning of 2009?
Instantaneous Rate of Change at t = 9
𝑑𝑃 𝑑𝑡 = 𝑑 𝑑𝑡 𝑡

46. Applications Example 1
Population of Nevada, P, in millions can be approximated by 𝑃= 𝑡
Where t is years since the start of 2000.
At what rate was the population growing at the beginning of 2009?
Instantaneous Rate of Change at t = 9
𝑑𝑃 𝑑𝑡 = 𝑑 𝑑𝑡 𝑡 =2020 𝑑 𝑑𝑡 𝑡

47. Applications Example 1
Population of Nevada, P, in millions can be approximated by 𝑃= 𝑡
Where t is years since the start of 2000.
At what rate was the population growing at the beginning of 2009?
Instantaneous Rate of Change at t = 9
𝑑𝑃 𝑑𝑡 = 𝑑 𝑑𝑡 𝑡 =2020 𝑑 𝑑𝑡 𝑡 =2020 ln⁡(1.036) 𝑡

48. Applications Example 1
Population of Nevada, P, in millions can be approximated by 𝑃= 𝑡
Where t is years since the start of 2000.
At what rate was the population growing at the beginning of 2009?
Instantaneous Rate of Change at t = 9
𝑑𝑃 𝑑𝑡 = 𝑑 𝑑𝑡 𝑡 =2020 𝑑 𝑑𝑡 𝑡 =2020 ln⁡(1.036) 𝑡

49. Applications Example 1
Population of Nevada, P, in millions can be approximated by 𝑃= 𝑡
Where t is years since the start of 2000.
At what rate was the population growing at the beginning of 2009?
Instantaneous Rate of Change at t = 9
𝑑𝑃 𝑑𝑡 = 𝑑 𝑑𝑡 𝑡 =2020 𝑑 𝑑𝑡 𝑡 =2020 ln =98.22

50. Applications Example 1
Population of Nevada, P, in millions can be approximated by 𝑃= 𝑡
Where t is years since the start of 2000.
At what rate was the population growing at the beginning of 2009?
Solution million people/year
𝑑𝑃 𝑑𝑡 = 𝑑 𝑑𝑡 𝑡 =2020 𝑑 𝑑𝑡 𝑡 =2020 ln =98.22

51. Applications Example 2
At a time t hours after it was administered, the concentration of a drug in the body is 𝑓 𝑡 =27 𝑒 −0.14𝑡 ng/ml (nanograms/milliliter).
What is the concentration 4 hours after it was administered?
At what rate is the concentration changing at that time?

52. Applications Example 2
At a time t hours after it was administered, the concentration of a drug in the body is 𝑓 𝑡 =27 𝑒 −0.14𝑡 ng/ml (nanograms/milliliter).
What is the concentration 4 hours after it was administered?
𝑓 4 =27 𝑒 −0.14(4) =15.42 𝑛𝑔/𝑚𝑙

53. Applications Example 2
At a time t hours after it was administered, the concentration of a drug in the body is 𝑓 𝑡 =27 𝑒 −0.14𝑡 ng/ml (nanograms/milliliter).
At what rate is the concentration changing at that time?
𝑓′ 𝑡 =27 (−0.14)𝑒 −0.14𝑡

54. Applications Example 2
At a time t hours after it was administered, the concentration of a drug in the body is 𝑓 𝑡 =27 𝑒 −0.14𝑡 ng/ml (nanograms/milliliter).
At what rate is the concentration changing at that time?
𝑓 ′ (4)=27 −0.14 𝑒 − =−2.16 𝑛𝑔 𝑚𝑙 𝑝𝑒𝑟 ℎ𝑜𝑢𝑟