Introduction Derivatives Notation Application
Math 3A: Calculus with Applications
Lecture 07: The Derivative
• A Formal Definition of the Derivative„
• A Bunch of Example Derivatives,
• An Application & More!
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Lecture 07: The Derivative Peter Garfield, UCSB Mathematics
Introduction Derivatives Notation Application
Formal Definition of Derivative
If y = f(x) is a function, then
f′ (a) = the derivative of f(x) at x = a
= the slope of the tangent line to y = f(x)
at the point (x, y) = (a, f(a))
dy
=
dx x=a
How Do We Compute The Derivative?
f′ (a) = the limit of slopes of secant lines to y = f(x)
between (x, y) = (a, f(a)) and (x, f(x))
Lecture 07: The Derivative Peter Garfield, UCSB Mathematics
Introduction Derivatives Notation Application
Derivative as a limit
y
f(x)
f(x)−f(a)
f(a)
x−a
rise f(x)−f(a)
m= run = x−a
a x x
Lecture 07: The Derivative Peter Garfield, UCSB Mathematics
Introduction Derivatives Notation Application
Let x approach a
y
rise f(x)−f(a)
m= run = x−a
f(x)−f(a)
slope of tangent = lim x−a
x→a
a x x
Lecture 07: The Derivative Peter Garfield, UCSB Mathematics
Introduction Derivatives Notation Application
Find the derivatives
f(x) − f(a)
1. Use the definition of derivative f′ (a) = lim to find the derivative of
x→a x−a
f(x) = 2x + 7 at x = 1.
(A) 0 (B) 1 (C) 2 (D) 3 (E) DNE
y
f(x) − f(1) (2x + 7) − (9)
=
12 x−1 x−1
2x − 2
=
x−1
8
2(x − 1)
=
x−1
4 Answer: C
= 2 if x ̸= 1
f(x) − f(1)
x lim = lim(2) = 2.
1 2 3 x→1 x−1 x→1
Lecture 07: The Derivative Peter Garfield, UCSB Mathematics
Introduction Derivatives Notation Application
Find the derivatives
f(x) − f(a)
2. Use the definition of derivative f′ (a) = lim to find the derivative of
x→a x−a
f(x) = x2 at x = 2.
(A) 1 (B) 2 (C) 3 (D) 4 (E) DNE
12 f(x) − f(2) x2 − 4
=
x−2 x−2
Answer: D (x + 2)(x − 2)
8 =
x−2
4 = x + 2 if x ̸= 2
f(x) − f(2)
lim = lim(x + 2) = 4.
x→2 x−2 x→2
1 2 3 x
Lecture 07: The Derivative Peter Garfield, UCSB Mathematics
Introduction Derivatives Notation Application
Find the derivatives
f(x) − f(a)
3. Use the definition of derivative f′ (a) = lim to find the derivative of
x→a x−a
f(x) = x3 + x at x = 1.
(A) 1 (B) 2 (C) 3 (D) 4 (E) DNE
12 f(x) − f(1) x3 + x − 2
=
x−1 x−1
8 (x + x + 2)(x − 1)
2
=
x−1
4 = x2 + x + 2 if x ̸= 1
Answer: D f(x) − f(1)
lim = lim(x2 + x + 2) = 4.
x→1 x−1 x→1
1 2 3 x
Lecture 07: The Derivative Peter Garfield, UCSB Mathematics
Introduction Derivatives Notation Application
Find the derivatives
f(x) − f(a)
4. Use the definition of derivative f′ (a) = lim to find the derivative of
√ x→a x−a
f(x) = x at x = 4.
(A) 0 (B) 1/4 (C) 1/2 (D) 1 (E) DNE
√ √
f(x) − f(4) x−2 x+2
3y = ·√
x−4 x−4 x+2
2 x−4
= √
(x − 4)( x + 2)
1 1
Answer: B =√ if x ̸= 4
x+2
0 1 2 3 4 5 x f(x) − f(4) 1 1
lim = lim √ = .
x→4 x−4 x→4 x+2 4
Lecture 07: The Derivative Peter Garfield, UCSB Mathematics
Introduction Derivatives Notation Application
Find the derivatives
f(x) − f(a)
5. Use the definition of derivative f′ (a) = lim to find the derivative of
x→a x−a
f(x) = |x| at x = 0.
(A) −1 (B) 0 (C) 1 (D) 2 (E) DNE
y
f(x) − f(0) |x| − 0
=
1 x−0 x−0
|x|
Answer: E =
x
.
f(x) − f(0)
−1 0 1 x lim DNE
x→4 x−0
Lecture 07: The Derivative Peter Garfield, UCSB Mathematics
Introduction Derivatives Notation Application
Alternate Notation
y
f(x)f(a+h)
f(x)−f(a)f(a+h)−f(a)
f(a)
x−ax−a=h
rise f(a +h)
f(a+ h)−−f(a)
f(a)
f′ (a)
m = = lim =
runh→0 hh
a xa+h x
Lecture 07: The Derivative Peter Garfield, UCSB Mathematics
Introduction Derivatives Notation Application
Question
6. We can think of the limit
(1 + h)−2 − 1
lim
h→0 h
as a derivative f′ (a) for some function f(x) and some value x = a. If we
assume a = 1, then this is f′ (1) for some function f(x). Which function?
(A) f(x) = x2 (B) f(x) = x−2 (C) f(x) = x−2 − 1
(D) f(x) = (x + h)−2 (E) None of these
Idea: f(1) = 1 and f(1 + h) = (1 + h)−2 . So f ( ) = −2
.
Answer: B
Lecture 07: The Derivative Peter Garfield, UCSB Mathematics
Introduction Derivatives Notation Application
Wolfe’s Royalties
7. For his 1936 memoir The Story of a Novel, Thomas Wolfe received royalties
(in dollars) of
150x if 0 ≤ x ≤ 3
R(x) = 450 + 187.5(x − 3) if 3 < x ≤ 7.5
1293.75 + 225(x − 7.5) if x > 7.5
where x represents the number of thousands of books sold.
(A) Roughly sketch this function.
(B) Using the limit definition of derivative, try to compute R′ (1). In your own
words, what does R′ (1) represent? How does it relate to your sketch?
(C) Using the limit definition of derivative, try to compute R′ (3). Does it exist?
Why or why not? How does your answer relate to your sketch?
Lecture 07: The Derivative Peter Garfield, UCSB Mathematics
Introduction Derivatives Notation Application
Wolfe
y
Royalties II
m=225
$1, 293.75
$1000
m=187.5
$450
m=150
3 6 7.5 9 12 x
Lecture 07: The Derivative Peter Garfield, UCSB Mathematics
