1. Section 2.2
The Derivative as a Function
V63.0121.002.2010Su, Calculus I
New York University
May 24, 2010
Announcements
Homework 1 due Tuesday
Quiz 2 Thursday in class on Sections 1.5–2.5
. . . . . .

2. Announcements
Homework 1 due Tuesday
Quiz 2 Thursday in class
on Sections 1.5–2.5
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 2 / 28

3. Objectives
Given a function f, use the
definition of the derivative
to find the derivative
function f’.
Given a function, find its
second derivative.
Given the graph of a
function, sketch the graph
of its derivative.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 3 / 28

4. Derivative
. . . . . .

5. Recall: the derivative
Definition
Let f be a function and a a point in the domain of f. If the limit
f(a + h) − f(a) f(x) − f(a)
f′ (a) = lim = lim
h→0 h x→a x−a
exists, the function is said to be differentiable at a and f′ (a) is the
derivative of f at a.
The derivative …
…measures the slope of the line through (a, f(a)) tangent to the
curve y = f(x);
…represents the instantaneous rate of change of f at a
…produces the best possible linear approximation to f near a.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 4 / 28

6. Derivative of the reciprocal function
Example
1
Suppose f(x) = . Use the
x
definition of the derivative to
find f′ (2).
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 6 / 28

7. Derivative of the reciprocal function
Example
1
Suppose f(x) = . Use the
x
definition of the derivative to x
.
find f′ (2).
Solution
.
′ 1/x − 1/2 2−x .
f (2) = lim = lim x
.
x→2 x−2 x→2 2x(x − 2)
−1 1
= lim =−
x→2 2x 4
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 6 / 28

8. Outline
What does f tell you about f′ ?
How can a function fail to be differentiable?
Other notations
The second derivative
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 7 / 28

9. 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′ ?
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 8 / 28

10. 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 (technically,
nonpositive) on that interval
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 8 / 28

11. Derivative of the reciprocal function
Example
1
Suppose f(x) = . Use the
x
definition of the derivative to x
.
find f′ (2).
Solution
.
′ 1/x − 1/2 2−x .
f (2) = lim = lim x
.
x→2 x−2 x→2 2x(x − 2)
−1 1
= lim =−
x→2 2x 4
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 9 / 28

12. 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 (technically,
nonpositive) on that interval
If f is increasing on an interval, f′ is positive (technically,
nonnegative) on that interval
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 10 / 28

13. Graphically and numerically
y
.
x2 − 22
x m=
x−2
3 5
. .
9 .
2.5 4.5
2.1 4.1
2.01 4.01
. .25 .
6 .
limit 4
. .41 .
4 . 1.99 3.99
. .0401 .
4.9601 .
3 . .61
3
4
. .. 1.9 3.9
1.5 3.5
. .25 .
2 .
1 3
. .
1 .
. . . ... . . x
.
1 1 . .. .1 3
. . .5 .99 .5 .
12 .
2.9 2
2
.01
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 11 / 28

14. What does f tell you about f′ ?
Fact
If f is decreasing on (a, b), then f′ ≤ 0 on (a, b).
Proof.
If f is decreasing on (a, b), and ∆x > 0, then
f(x + ∆x) − f(x)
f(x + ∆x) < f(x) =⇒ <0
∆x
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 12 / 28

15. What does f tell you about f′ ?
Fact
If f is decreasing on (a, b), then f′ ≤ 0 on (a, b).
Proof.
If f is decreasing on (a, b), and ∆x > 0, then
f(x + ∆x) − f(x)
f(x + ∆x) < f(x) =⇒ <0
∆x
But if ∆x < 0, then x + ∆x < x, and
f(x + ∆x) − f(x)
f(x + ∆x) > f(x) =⇒ <0
∆x
still!
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 12 / 28

16. What does f tell you about f′ ?
Fact
If f is decreasing on (a, b), then f′ ≤ 0 on (a, b).
Proof.
If f is decreasing on (a, b), and ∆x > 0, then
f(x + ∆x) − f(x)
f(x + ∆x) < f(x) =⇒ <0
∆x
But if ∆x < 0, then x + ∆x < x, and
f(x + ∆x) − f(x)
f(x + ∆x) > f(x) =⇒ <0
∆x
still! Either way,
f(x + ∆x) − f(x) f(x + ∆x) − f(x)
< 0 =⇒ f′ (x) = lim ≤0
∆x ∆x→0 ∆x . . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 12 / 28

17. Another important derivative fact
Fact
If the graph of f has a horizontal tangent line at c, then f′ (c) = 0.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 13 / 28

18. Another important derivative fact
Fact
If the graph of f has a horizontal tangent line at c, then f′ (c) = 0.
Proof.
The tangent line has slope f′ (c). If the tangent line is horizontal, its
slope is zero.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 13 / 28

19. Outline
What does f tell you about f′ ?
How can a function fail to be differentiable?
Other notations
The second derivative
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 14 / 28

20. Differentiability is super-continuity
Theorem
If f is differentiable at a, then f is continuous at a.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 15 / 28

21. 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 · lim (x − a)
x→a x−a x→a
′
= f (a) · 0 = 0
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 15 / 28

22. 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 · lim (x − a)
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.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 15 / 28

23. Differentiability FAIL
Kinks
f
.(x)
. x
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 16 / 28

24. Differentiability FAIL
Kinks
f
.(x) .′ (x)
f
. x
. . x
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 16 / 28

25. Differentiability FAIL
Kinks
f
.(x) .′ (x)
f
.
. x
. . x
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 16 / 28

26. Differentiability FAIL
Cusps
f
.(x)
. x
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 17 / 28

27. Differentiability FAIL
Cusps
f
.(x) .′ (x)
f
. x
. . x
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 17 / 28

28. Differentiability FAIL
Cusps
f
.(x) .′ (x)
f
. x
. . x
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 17 / 28

29. Differentiability FAIL
Vertical Tangents
f
.(x)
. x
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 18 / 28

30. Differentiability FAIL
Vertical Tangents
f
.(x) .′ (x)
f
. x
. . x
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 18 / 28

31. Differentiability FAIL
Vertical Tangents
f
.(x) .′ (x)
f
. x
. . x
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 18 / 28

32. Differentiability FAIL
Weird, Wild, Stuff
f
.(x)
. x
.
This function is differentiable at
0.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 19 / 28

33. Differentiability FAIL
Weird, Wild, Stuff
f
.(x) .′ (x)
f
. x
. . x
.
This function is differentiable at But the derivative is not
0. continuous at 0!
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 19 / 28

34. Outline
What does f tell you about f′ ?
How can a function fail to be differentiable?
Other notations
The second derivative
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 20 / 28

35. Notation
Newtonian notation
f′ (x) y′ (x) y′
Leibnizian notation
dy d df
f(x)
dx dx dx
These all mean the same thing.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 21 / 28

36. Link between the notations
f(x + ∆x) − f(x) ∆y dy
f′ (x) = lim = lim =
∆x→0 ∆x ∆x→0 ∆x dx
dy
Leibniz thought of as a quotient of “infinitesimals”
dx
dy
We think of as representing a limit of (finite) difference
dx
quotients, not as an actual fraction itself.
The notation suggests things which are true even though they
don’t follow from the notation per se
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 22 / 28

37. Meet the Mathematician: Isaac Newton
English, 1643–1727
Professor at Cambridge
(England)
Philosophiae Naturalis
Principia Mathematica
published 1687
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 23 / 28

38. Meet the Mathematician: Gottfried Leibniz
German, 1646–1716
Eminent philosopher as
well as mathematician
Contemporarily disgraced
by the calculus priority
dispute
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 24 / 28

39. Outline
What does f tell you about f′ ?
How can a function fail to be differentiable?
Other notations
The second derivative
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 25 / 28

40. 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!
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 26 / 28

41. 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:
d2 y d2 d2 f
f(x)
dx2 dx2 dx2
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 26 / 28

42. function, derivative, second derivative
y
.
.(x) = x2
f
.′ (x) = 2x
f
.′′ (x) = 2
f
. x
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 27 / 28

43. Summary
A function can be differentiated at every point to find its derivative
function.
The derivative of a function notices the monotonicity of the
function (fincreasing =⇒ f′ ≥ 0)
The second derivative of a function measures the rate of the
change of the rate of change of that function.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.2 The Derivative as a Function May 24, 2010 28 / 28