The derivative is one of the fundamental quantities in calculus, partly because it is ubiquitous in nature. We give examples of it coming about, a few calculations, and ways information about the function an imply information about the derivative

1. Sections 2.1–2.2
Derivatives and Rates of Changes
The Derivative as a Function
V63.0121, Calculus I
February 9–12, 2009
Announcements
Quiz 2 is next week: Covers up through 1.6
Midterm is March 4/5: Covers up to 2.4 (next T/W)

2. Outline
Rates of Change
Tangent Lines
Velocity
Population growth
Marginal costs
The derivative, defined
Derivatives of (some) power functions
What does f tell you about f ?
How can a function fail to be differentiable?
Other notations
The second derivative

3. The tangent problem
Problem
Given a curve and a point on the curve, find the slope of the line
tangent to the curve at that point.

4. The tangent problem
Problem
Given a curve and a point on the curve, find the slope of the line
tangent to the curve at that point.
Example
Find the slope of the line tangent to the curve y = x 2 at the point
(2, 4).

5. Graphically and numerically
y
x m
4
x
2

6. Graphically and numerically
y
x m
3 5
9
4
x
2 3

7. Graphically and numerically
y
x m
3 5
2.5 4.25
6.25
4
x
2 2.5

8. Graphically and numerically
y
x m
3 5
2.5 4.25
2.1 4.1
4.41
4
x
2.1
2

9. Graphically and numerically
y
x m
3 5
2.5 4.25
2.1 4.1
2.01 4.01
4.0401
4
x
2.01
2

10. Graphically and numerically
y
x m
3 5
2.5 4.25
2.1 4.1
2.01 4.01
4
1 3
1
x
1 2

11. Graphically and numerically
y
x m
3 5
2.5 4.25
2.1 4.1
2.01 4.01
4
1.5 3.5
2.25 1 3
x
1.5 2

12. Graphically and numerically
y
x m
3 5
2.5 4.25
2.1 4.1
2.01 4.01
1.9 3.9
4
3.61
1.5 3.5
1 3
x
1.9
2

13. Graphically and numerically
y
x m
3 5
2.5 4.25
2.1 4.1
2.01 4.01
1.99 3.99
1.9 3.9
4
3.9601
1.5 3.5
1 3
x
1.99
2

14. Graphically and numerically
y
x m
3 5
2.5 4.25
9
2.1 4.1
2.01 4.01
6.25 limit 4
1.99 3.99
4.41 1.9 3.9
4.0401
4
3.9601
3.61
1.5 3.5
2.25 1 3
1
x
1 1.5 2.1 3
1.99
1.9 2.5
2.01
2

15. The tangent problem
Problem
Given a curve and a point on the curve, find the slope of the line
tangent to the curve at that point.
Example
Find the slope of the line tangent to the curve y = x 2 at the point
(2, 4).
Upshot
If the curve is given by y = f (x), and the point on the curve is
(a, f (a)), then the slope of the tangent line is given by
f (x) − f (a)
mtangent = lim
x −a
x→a

16. Velocity
Problem
Given the position function of a moving object, find the velocity of
the object at a certain instant in time.
Example
Drop a ball off the roof of the Silver Center so that its height can
be described by
h(t) = 50 − 10t 2
where t is seconds after dropping it and h is meters above the
ground. How fast is it falling one second after we drop it?

17. Velocity
Problem
Given the position function of a moving object, find the velocity of
the object at a certain instant in time.
Example
Drop a ball off the roof of the Silver Center so that its height can
be described by
h(t) = 50 − 10t 2
where t is seconds after dropping it and h is meters above the
ground. How fast is it falling one second after we drop it?
Solution
The answer is
(50 − 10t 2 ) − 40
= −20.
lim
t −1
t→1

18. Numerical evidence
h(t) − h(1)
t vave =
t −1
−30
2
−25
1.5
−21
1.1
−20.01
1.01
−20.001
1.001

19. Velocity
Problem
Given the position function of a moving object, find the velocity of
the object at a certain instant in time.
Example
Drop a ball off the roof of the Silver Center so that its height can
be described by
h(t) = 50 − 10t 2
where t is seconds after dropping it and h is meters above the
ground. How fast is it falling one second after we drop it?
Solution
The answer is
(50 − 10t 2 ) − 40
= −20.
lim
t −1
t→1

20. Upshot
If the height function is given by h(t), the instantaneous velocity
at time t is given by
h(t + ∆t) − h(t)
v = lim
∆t
∆t→0

21. Population growth
Problem
Given the population function of a group of organisms, find the
rate of growth of the population at a particular instant.

22. Population growth
Problem
Given the population function of a group of organisms, find the
rate of growth of the population at a particular instant.
Example
Suppose the population of fish in the East River is given by the
function
3e t
P(t) =
1 + et
where t is in years since 2000 and P is in millions of fish. Is the
fish population growing fastest in 1990, 2000, or 2010? (Estimate
numerically)?

23. Numerical evidence
P(−10 + 0.1) − P(−10)
r1990 ≈ ≈ 0.000136
0.1

24. Numerical evidence
P(−10 + 0.1) − P(−10)
r1990 ≈ ≈ 0.000136
0.1
P(0.1) − P(0)
r2000 ≈ ≈ 0.75
0.1

25. Numerical evidence
P(−10 + 0.1) − P(−10)
r1990 ≈ ≈ 0.000136
0.1
P(0.1) − P(0)
r2000 ≈ ≈ 0.75
0.1
P(10 + 0.1) − P(10)
r2010 ≈ ≈ 0.000136
0.1

26. Population growth
Problem
Given the population function of a group of organisms, find the
rate of growth of the population at a particular instant.
Example
Suppose the population of fish in the East River is given by the
function
3e t
P(t) =
1 + et
where t is in years since 2000 and P is in millions of fish. Is the
fish population growing fastest in 1990, 2000, or 2010? (Estimate
numerically)?
Solution
The estimated rates of growth are 0.000136, 0.75, and 0.000136.

27. Upshot
The instantaneous population growth is given by
P(t + ∆t) − P(t)
lim
∆t
∆t→0

28. Marginal costs
Problem
Given the production cost of a good, find the marginal cost of
production after having produced a certain quantity.

29. Marginal costs
Problem
Given the production cost of a good, find the marginal cost of
production after having produced a certain quantity.
Example
Suppose the cost of producing q tons of rice on our paddy in a
year is
C (q) = q 3 − 12q 2 + 60q
We are currently producing 5 tons a year. Should we change that?

30. Comparisons
q C (q) AC (q) = C (q)/q ∆C = C (q + 1) − C (q)
4 112 28 13
5 125 25 19
6 144 24 31

31. Marginal costs
Problem
Given the production cost of a good, find the marginal cost of
production after having produced a certain quantity.
Example
Suppose the cost of producing q tons of rice on our paddy in a
year is
C (q) = q 3 − 12q 2 + 60q
We are currently producing 5 tons a year. Should we change that?
Example
If q = 5, then C = 125, ∆C = 19, while AC = 25. So we should
produce more to lower average costs.

32. Upshot
The incremental cost
∆C = C (q + 1) − C (q)
is useful, but depends on units.

33. Upshot
The incremental cost
∆C = C (q + 1) − C (q)
is useful, but depends on units.
The marginal cost after producing q given by
C (q + ∆q) − C (q)
MC = lim
∆q
∆q→0
is more useful since it’s unit-independent.

34. Outline
Rates of Change
Tangent Lines
Velocity
Population growth
Marginal costs
The derivative, defined
Derivatives of (some) power functions
What does f tell you about f ?
How can a function fail to be differentiable?
Other notations
The second derivative

35. The definition
All of these rates of change are found the same way!

36. The definition
All of these rates of change are found the same way!
Definition
Let f be a function and a a point in the domain of f . If the limit
f (a + h) − f (a)
f (a) = lim
h
h→0
exists, the function is said to be differentiable at a and f (a) is
the derivative of f at a.

37. Derivative of the squaring function
Example
Suppose f (x) = x 2 . Use the definition of derivative to find f (a).

38. Derivative of the squaring function
Example
Suppose f (x) = x 2 . Use the definition of derivative to find f (a).
Solution
(a + h)2 − a2
f (a + h) − f (a)
f (a) = lim = lim
h h
h→0 h→0
2 + 2ah + h2 ) − a2 2ah + h2
(a
= lim = lim
h h
h→0 h→0
= lim (2a + h) = 2a.
h→0

39. What does f tell you about f ?
If f is a function, we can compute the derivative f (x) at each
point x where f is differentiable, and come up with another
function, the derivative function.
What can we say about this function f ?
If f is decreasing on an interval, f is negative (well,
nonpositive) on that interval
If f is increasing on an interval, f is positive (well,
nonnegative) on that interval

40. Outline
Rates of Change
Tangent Lines
Velocity
Population growth
Marginal costs
The derivative, defined
Derivatives of (some) power functions
What does f tell you about f ?
How can a function fail to be differentiable?
Other notations
The second derivative

41. Differentiability is super-continuity
Theorem
If f is differentiable at a, then f is continuous at a.

42. Differentiability is super-continuity
Theorem
If f is differentiable at a, then f is continuous at a.
Proof.
We have
f (x) − f (a)
lim (f (x) − f (a)) = lim · (x − a)
x −a
x→a x→a
f (x) − f (a)
· lim (x − a)
= lim
x −a
x→a x→a
= f (a) · 0 = 0

43. Differentiability is super-continuity
Theorem
If f is differentiable at a, then f is continuous at a.
Proof.
We have
f (x) − f (a)
lim (f (x) − f (a)) = lim · (x − a)
x −a
x→a x→a
f (x) − f (a)
· lim (x − a)
= lim
x −a
x→a x→a
= f (a) · 0 = 0
Note the proper use of the limit law: if the factors each have a
limit at a, the limit of the product is the product of the limits.

44. How can a function fail to be differentiable?
Kinks
f (x)
x

45. How can a function fail to be differentiable?
Kinks
f (x) f (x)
x x

46. How can a function fail to be differentiable?
Kinks
f (x) f (x)
x x

47. How can a function fail to be differentiable?
Cusps
f (x)
x

48. How can a function fail to be differentiable?
Cusps
f (x) f (x)
x x

49. How can a function fail to be differentiable?
Cusps
f (x) f (x)
x x

50. How can a function fail to be differentiable?
Vertical Tangents
f (x)
x

51. How can a function fail to be differentiable?
Vertical Tangents
f (x) f (x)
x x

52. How can a function fail to be differentiable?
Vertical Tangents
f (x) f (x)
x x

53. How can a function fail to be differentiable?
Weird, Wild, Stuff
f (x)
x

54. How can a function fail to be differentiable?
Weird, Wild, Stuff
f (x) f (x)
x x

55. Outline
Rates of Change
Tangent Lines
Velocity
Population growth
Marginal costs
The derivative, defined
Derivatives of (some) power functions
What does f tell you about f ?
How can a function fail to be differentiable?
Other notations
The second derivative

56. Notation
Newtonian notation
f (x) y (x) y
Leibnizian notation
dy d df
f (x)
dx dx dx
These all mean the same thing.

57. Meet the Mathematician: Isaac Newton
English, 1643–1727
Professor at Cambridge
(England)
Philosophiae Naturalis
Principia Mathematica
published 1687

58. Meet the Mathematician: Gottfried Leibniz
German, 1646–1716
Eminent philosopher as
well as mathematician
Contemporarily disgraced
by the calculus priority
dispute

59. Outline
Rates of Change
Tangent Lines
Velocity
Population growth
Marginal costs
The derivative, defined
Derivatives of (some) power functions
What does f tell you about f ?
How can a function fail to be differentiable?
Other notations
The second derivative

60. The second derivative
If f is a function, so is f , and we can seek its derivative.
f = (f )
It measures the rate of change of the rate of change!

61. The second derivative
If f is a function, so is f , and we can seek its derivative.
f = (f )
It measures the rate of change of the rate of change! Leibnizian
notation:
d 2y d2 d 2f
f (x)
dx 2 dx 2 dx 2

62. function, derivative, second derivative
y
f (x) = x 2
f (x) = 2x
f (x) = 2
x