Slide 1
Calculus I Announcements
• Office Hours: Amos Eaton 309, Mondays 12:50-2:50
• Exam 2 is Thursday, October 22nd. The study guide
is now on the course web page. Start studying now,
and make a plan to succeed.
• Read sections 3.2,3.3, and 3.6. (We are skipping 3.4,
but we will cover 3.5 next lecture.)
• Do the homework for sections as follows
– 3.2, (Note that in Section 3.2, you need to use the
definition on some of the questions, as per the
instructions.)
– 3.3, do all of it
– 3.6, do up through 31 on the study guide, the last
4 questions on the study guide from 3.6 can be
postponed until after Thursday’s lecture.
Slide 2
Derivatives, Overview
It is important to know at least 3 aspects of the derivative:
• The Definition
Recall that the definition of derivative of a function f
with respect to a variable x is
f (x + h) − f (x)
f 0 (x) = lim
h→0
h
• The Rules
Since derivatives are used so often, rules are created to
make them easier to find. Here is a list of names of a
few of the rules we will learn
– Power rule, sum rule, constant multiple rule,
product rule, quotient rule, rule for exponentials,
rules for trigonometric functions, chain rule, rules
for logarithms, rules for inverse functions.
• Interpretations/Applications
– The derivative of a function gives the slope of the
tangent line to the function.
– The derivative of a function gives the
(instantaneous) rate of change of the function.
– The derivative of a population with respect to
time gives the rate of change of the population.
– The derivative of position with respect to time
gives instantaneous velocity
– The derivative of velocity with respect to time
gives acceleration
– The derivative of mass with respect to distance
gives linear density
– The derivative of charge with respect to time gives
current
– Many, many other applications...
Slide 3
Derivatives
Likely Exam Essay Question: State the definition of
derivative. Illustrate the definition with a picture and
explain how the derivative gives the instantaneous slope of
the graph.
Slide 4
Derivatives: Notation
You need to know different notations for the derivative of
a function y = f (x).
dy
df
d
=
=
f (x) = D(f )(x) = Dx f (x)
dx dx dx
All of these are equal to
f 0 (x) = y 0 =
f (x + h) − f (x)
h→0
h
f 0 (x) = lim
Also, for the derivative of the function f (x) at a particular
point x = a, we use all the following notations
dy df d
0
f (a) =
=
=
f
(x)
dx x=a dx x=a dx
x=a
Slide 5
Sketching Derivatives
The derivative of a function is another function.
1. Piecewise linear functions
• The derivative of a straight line y = mx + b is a
constant function y 0 = m
• Given the graph of a piecewise linear function,
sketch the graph of the derivative.
2. Smooth functions
• The derivative of a general function gives its slope
at each point (i.e. the slope of the tangent line).
• Given the graph of a general function, sketch the
graph of the derivative.
Slide 6
Differentiability
A function is differentiable at x = a if lim
h→0
exists.
f (a + h) − f (a)
h
• If a function is differentiable at x = a then the
function is continuous at x = a
• But the previous statement is a one-way-street:
– a function can be continuous at x = a, but not
differentiable at x = a.
Draw a graph to illustrate this.
• For the graph of a given function, identify where it
appears to be discontinuous and where it appears to
be non-differentiable.
Slide 7
Derivative Rules
f (x + h) − f (x)
h→0
h
Recall f 0 (x) = lim
1. Constant Rule
d
c = 0 when c is constant
dx
d n
x = nxn−1 (for any real number n 6= 0)
dx
d
3. Constant multiple rule
cf (x) = cf 0 (x)
dx
d
4. Sum rule
(f (x) + g(x)) = f 0 (x) + g 0 (x)
dx
2. Power Rule
√ d
x4 + 3x2 − x
dx
d x5 + x2
• Use the rules to find
dx
x3
• Derive the constant rule
• Use the rules to find
• Derive the power rule for integers n = 2, 3
Slide 8
iClicker
Question Find y 0 if y = 4x3 + 3x + ln(7)
A. 12x2 + 3 + 1/7
B. 12x2 + 3
C. 4x2 + 3 + 1/7
D. 4x2 + 3
E. None of the above
Answer to Question Find y 0 if y = 4x3 + 3x + ln(7)
A. 12x2 + 3 + 1/7
B. 12x2 + 3
is the correct answer.
C. 4x2 + 3 + 1/7
D. 4x2 + 3
E. None of the above
Slide 9
Derivative Rules
Here’s a list of a few of the standard derivatives to
memorize. We will derive some of these later.
d
c = 0 for any constant c
1.
dx
d
2.
x=1
dx
d r
3.
x = rxr−1 for r 6= 0
dx
d x
e = ex
4.
dx
d x
5.
a = ax ln(a) for a > 0
dx
d
6.
ln(x) = 1/x
dx
d
sin(x) = cos(x)
7.
dx
d
8.
cos(x) = − sin(x)
dx
d
9.
tan(x) = sec2 (x)
dx
d
10.
cot(x) = − csc2 (x)
dx
d
11.
sec(x) = sec(x) tan(x)
dx
d
12.
csc(x) = − csc(x) cot(x)
dx
Slide 10
iClicker
Question
Find f 0 (x) if f (x) = 3x
A. 3x+1 /(x + 1)
B. x3x−1
C. 3x
D. 3x ln(3)
E. None of the above
Answer to Question
Find f 0 (x) if f (x) = 3x
A. 3x+1 /(x + 1)
B. x3x−1
C. 3x
D. 3x ln(3)
is the correct answer.
E. None of the above
Slide 11
Question
iClicker
Find f 0 (x) if f (x) = sec(x) + cot(x)
A. tan2 (x) − sec2 (x)
B. tan2 (x) − csc2 (x)
C. sec(x) tan(x) − sec2 (x)
D. sec(x) tan(x) − csc2 (x)
E. None of the above
Answer to Question
f (x) = sec(x) + cot(x)
Find f 0 (x) if
A. tan2 (x) − sec2 (x)
B. tan2 (x) − csc2 (x)
C. sec(x) tan(x) − sec2 (x)
D. sec(x) tan(x) − csc2 (x)
E. None of the above
is the correct answer.
Slide 12
Derivative Rules
If f and g are differentiable functions,
• the Product Rule for u = f (x) and v = g(x) says
d
(f (x) · g(x)) = f 0 (x)g(x) + f (x)g 0 (x)
dx
or
du
dv
d
uv =
v+u
dx
dx
dx
1. Use the product rule to find the following derivatives:
d 2
(x + 3x + 5) · (x5 − 3x + 2)
dx
d x
e sin(x)
dx
d 3
(x + 3x + 4) cos(x)
dx
2. Derive the product rule (Possible lecture question on
exam 2, will be done in class on October 1.)
Slide 13
iClicker
Question Find
d
ln(x)(x5 + 3x − 2)
dx
5x4 + 3
A.
x
B. ln(x)(5x4 + 3)
x5 + 3x − 2
C. ln(x)(5x + 3) +
x
5x4 + 3
5
D. ln(x)(x + 3x − 2) +
x
E. None of the above
4
Answer to Question Find
d
ln(x)(x5 + 3x − 2)
dx
5x4 + 3
A.
x
B. ln(x)(5x4 + 3)
x5 + 3x − 2
C. ln(x)(5x + 3) +
x
answer.
5x4 + 3
D. ln(x)(x5 + 3x − 2) +
x
E. None of the above
4
is the correct
Slide 14
Derivative Rules
For differentiable functions f and g, (g(x) 6= 0)
• the Reciprocal Rule says
−g 0 (x)
d
1
=
dx g(x)
(g(x))2
• and the Quotient Rule says
g(x)f 0 (x) − f (x)g 0 (x)
d f (x)
=
dx g(x)
(g(x))2
1. Use the rules to find
d
1
dx sin(x)
d x2 + 3x + 2
dx ex + cos(x)
Slide 15
iClicker
d x2 + 3x
Question Find
dx 4x + 5
A.
4(x2 + 3x) − (2x + 3)(4x + 5)
(4x + 5)2
(4x + 5)(x2 + 3x) − 4(2x + 3)
B.
(4x + 5)2
(2x + 3)4 − (4x + 5)(x2 + 3x)
C.
(4x + 5)2
(2x + 3)(4x + 5) − 4(x2 + 3x)
D.
(4x + 5)2
E. None of the above
d x2 + 3x
Answer to Question Find
dx 4x + 5
4(x2 + 3x) − (2x + 3)(4x + 5)
A.
(4x + 5)2
(4x + 5)(x2 + 3x) − 4(2x + 3)
B.
(4x + 5)2
C.
(2x + 3)4 − (4x + 5)(x2 + 3x)
(4x + 5)2
(2x + 3)(4x + 5) − 4(x2 + 3x)
D.
(4x + 5)2
answer.
E. None of the above
is the correct