Chapter 3: Derivatives – Test Review
Calculus
page 1
Name_____________________________________
Chapter 3: Derivatives—Test Review, Part 1
Note: You will receive Part 2 later. There are answers at the end of each part.
Derivatives Review: Summary of Rules
Each derivative rule is summarized for you below. Write an example that would best exemplify each rule.
1. Power rule
f (x) = ax b then f '(x) = a× bx b -1
Example:
2. Constant rule
f (x) = c then f '(x) = 0
Example:
3. Product rule
y = f (x)× g(x) then y'= f (x)× g'(x) + g(x)× f '(x)
Example:
4. Quotient rule
y=
f (x)
g(x) × f '(x) - f (x) × g'(x)
then y'=
g(x)
[g(x)]2
Only use it when the denominator is a function of x; not for something like y =
Example:
5. Chain rule
y = f (g(x)) then y'= f '(g(x))× g'(x)
Or y = f (u) and u = g(x) then
Example:
dy dy du
=
×
or y'= f '(u) × g'(x)
dx du dx
2x - 5
3
Chapter 3: Derivatives – Test Review
Calculus
page 2
6. Logarithmic functions
y = ln x then y'=
1
x
more generally if y = ln u then
dy 1 du
= ×
dx u dx
Example:
7. Exponential functions
y = e x then y'= e x
more generally if y = e u then
dy
du
= eu ×
dx
dx
Example:
8. Trigonometric functions
(You need to memorize the derivatives below.)
d
du
d
du
sin u = cos u ×
cos u = - sin u ×
dx
dx
dx
dx
(You do NOT need to memorize the derivatives below.)
d
du
tan u = sec 2 u ×
dx
dx
d
du
cot u = - csc 2 u ×
dx
dx
Examples:
d
du
sec u = sec utan u ×
dx
dx
d
du
csc u = - csc ucot u
dx
dx
Chapter 3: Derivatives – Test Review
Calculus
page 3
Important Techniques for Differentiation
x 3 -1
1 3 -3
x2 - 3 x
1.
Simplify first!!! y =
 rewrite as y = - x 2  y'= + x 2
4 4
4 8
4x
( )
æ 1 ö
2
y = lnç ÷  rewrite as y = ln1- ln x = 0 - 2 ln( x) = 2 ln( x )
2
èx ø
2.
Implicit differentiation: when you cannot easily solve for y. Don’t forget to use the product rule
when necessary:
Example:
x 2 + 2xy + 3y 2 = 7
dy
dy
2x + 2x ×
+ 2y + 6y ×
=0
dx
dx
dy
dy
2x ×
+ 6y ×
= -2x - 2y
dx
dx
dy -2x - 2y - x - y
=
=
dx 2x + 6y
x + 3y
3.
Logarithmic differentiation: take the natural log of both sides, use the laws of logs to simplify,
and then use implicit differentiation to find the derivative.
Example:
y = 4 x × sin x
1
ln y = ln(4 x ) + ln( sin x ) = x ln 4 + ln(sin x)
2
1 dy
1 1
×
= ln 4 + ×
× cos x
y dx
2 sin x
dy
cos x ö
æ
= y × ln 4 +
è
dx
2sin x ø
dy
1
æ
ö
= 4 x sin x × ln 4 + cot x
è
ø
dx
2
Chapter 3: Derivatives – Test Review
Calculus
page 4
Derivative Review Questions – Complete on separate paper.
Find the derivative of the following functions. You might find it helpful to simplify before taking the
derivative. Simplify your answers.
2
2
1. g(t) = 2
2. h(x) =
3t
(3x) 2
3. y = x 3 - 5 +
3
x3
5. f (x) = x 3 (5 - 3x 2 )
7. f (x) =
x 2 + x -1
x2 -1
9. f (x) = 3 x 2 -1
11. g(t) =
t
(1- t) 3
13. y = 4e x
15. y =
2
x
e 2x
5
17. y =
1 + e 2x
19. y = ln
x(x - 1)
x-2
21. f (x) = ln 4x
23. y = ln
ex
1 + ex
4. f (x) =
x-
1
x
1ö
æ
6. s = 4 - 2 (t 2 - 3t)
è
t ø
8. h(x) =
4x2 + x
3x 2 - 2
10. h(x) =
2
x +1
12. f (t) = (t +1) t 2 +1
14. y = x 2e x
16. y = 3 xe 3x
ex
18. y =
1- xe x
20. y = x ln x
22. y = ln(x 2 - 2)
24. y = e -x sin x
1
x
25. y = cos3x
26. y = x 2 sin
27. y = cos2 x - sin 2 x
28. y =
29. y = sin x
30. y = sin5px
cos x
sin x
2
3
Chapter 3: Derivatives – Test Review
31. y =
cos x
x2
33. y = 3sin2 4 x + x
36.
32. y =
Calculus
page 5
cos(x - 1)
x -1
1
34. y = e sin 2x
2
35. f (x) =
(x - 4) 3 x 2
(3x + 1) 2
If f and g are the functions whose graphs are shown, let u(x) = f (x)× g(x) and v(x) =
f (x)
.
g(x)
Find:
y
4
a.
u’(1) = ________
b.
v’(5) = ________
f(x)
2
g(x)
c.
u’(0) = ________
5
x
-2
37.
a.
If g is a differentiable function, find an expression for the derivative of each of the following
functions.
æ 2x 4 + 3x ö
14g(x)
y=
b.
c.
y = lnç
y = x 6 × g(x)
÷
e 7x
è g(x) ø
38. Use implicit differentiation to find an equation of the (a.) tangent and (b.) normal line at the indicated
point.
at (-7, 3)
2y - y 3 = xy
dy
39. Find dx of these equations.
2
a. 3x y - 2x = cos(y)
5
3
b. 3x - 2y = p
y
3
2
c. e - 2x = xy
Chapter 3: Derivatives – Test Review
Calculus
page 6
40. Using the graphs below, calculate the following derivatives.
a.
h'(1) if h(x) = f (x) × g(x)
b.
j'(1) if j(x) =
c.
m'(1) if m(x) = f (g(x))
d.
k'(1) if k(x) = f ( f (x))
e.
n'(4) if n(x) = g( f (x))
f (x)
g(x)
41. Using the table below, calculate the following derivatives.
a) If h(x) = f (g(x)), find h¢(1).
b) If k(x) = g(x)× f (x), find k'(3).
c) If j(x) = [ f (x)] , find j'(2).
3
d) If m(x) =
f (x)
, find m'(3).
[ g(x)]2
x
1
2
3
f (x)
g(x)
f ¢(x)
g¢(x)
3
1
7
2
8
2
4
5
7
6
7
9
Chapter 3: Derivatives – Test Review
Calculus
page 7
ANSWERS to “Chapter 3: Derivatives — Test Review, Part 1”
-4
-4
9
1. g'(t) = 3
2. h'(x) = 3
3. y'= 3x 2 - 4
3t
9x
x
3
1
1
2
2
4. f '(x) =
5. f '(x) = 15x (1- x )
6. s'= 8t - 12 - 2
+
3
t
2 x 2 x
1
3
9
- x 2 - 16x - x 2
2x
x2 +1
2
7. f '(x) = - 2
8.
9. f '(x) =
2
2
2
2
2
(x - 1)
(3x - 2)
3(x - 1) 3
-1
2t 2 + t + 1
1 + 2t
10. h'(x) =
11. g'(t) =
12. f '(t) =
3
(1- t) 4
t2 +1
(x + 1) 2
1- 2x
2
13. y'= 8xe x
14. y'= xe x (x + 2)
15. y'= 2x
e
x
2x
x
1
e (x + 3)
-10e
e (1 + e x )
16. y'=
17. y'=
18. y'=
2
(1 + e 2x ) 2
(1- xe x ) 2
x 3
1
1
1
1
1
19. y'= +
20. y'= ln x +
21. f '(x) =
x x -1 x - 2
2x
2 ln x
x
4x
e
1
22. y'=
23. y'=124. y'= e -x (cos x - sin x)
=
2
x
x
3(x - 2)
1+ e
1+ e
1
1
25. y'= -3sin3x
26. y'= - cos + 2x sin
27. y'= -2sin2x or
x
x
28. y'= -csc 2 x or
y'= -1 - cot 2 x
31. y'=
cos x
29. y'=
2 sin x
- x sin x - 2cos x
x3
37 a. y'= x 6 × g'(x) + 6g(x)x 5
5
8
40. a. -6
b.
41. a. 30
b. 77
30. y'= 5p × cos(5px)
-(x -1)sin(x - 1) - cos(x - 1)
(x - 1) 2
34. y'= cos2x × e sin 2x
32. y'=
33. y'= 24sin(4x)cos(4x) +1 =12sin(8x) +1
(x - 4) 3 x 2 æ 3
2
6 ö
35. f '(x) =
+ 2
(3x + 1) è x - 4 x 3x + 1ø
-2
36 a. 0
b.
c. DNE
3
1
38 a. y = - (x + 7) + 3
6
2 - 6xy
39. a. y' = 2
3x + sin y
y'= -4 sin x cos x
b. y'=
8x 3 + 3 g'(x)
2x 4 + 3x g(x)
b. y = 6(x + 7) + 3
5x 4
b. y' = 2
2y
c. 4
d. -1
c. 15
d. -14
e.
-4
3
6x 2 + y 2
c. y' = y
e - 2xy
c. y'=
14 × g'(x) - 98g(x)
e 7x
Chapter 3: Derivatives – Test Review
Chapter 3: Derivatives — Test Review, Part 2
1. For the functions below:
I.
Where is it discontinuous? Name the
type.
II.
Where is it not differentiable? Name the
type.
a.
c.
d.
b.
Calculus
page 8
Chapter 3: Derivatives – Test Review
2. Sketch the graph of the derivatives of these
functions.
a.
b.
Calculus
page 9
Chapter 3: Derivatives – Test Review
c.
Calculus
page 10
3. The graph below shows the position of a
particle on a coordinate line.
(a) When does the particle change direction?
(b) When does the particle move at its greatest
speed?
(c) Graph the particle’s velocity. Think about the
following questions: When is the particle’s
velocity positive? negative? zero?
4. The graph below shows the velocity of a
particle moving on a coordinate line.
(a) When does the particle change direction?
(b) When does the particle move at its greatest
speed?
(c) When does the particle have its greatest
acceleration?
(d) Graph the particle’s acceleration. Think
about the following questions: When is the
particle’s acceleration positive? negative?
zero?
Chapter 3: Derivatives – Test Review
5. The following graph shows the velocity of a
skydiver after he jumps out of a plane. At
some point during his fall, his parachute
opens to slow his fall before he lands on the
ground. Here, positive velocity indicates a
downward speed.
Calculus
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7. The cost of extracting T tons of ore from a
copper mine is C = f (T ), where C is in
dollars. Suppose that f (1000) = 200,000 and
f ¢(1000) = 250.
a) What are the units of f ¢(T )?
b) What is the average cost per ton of
extracting 1,000 tons?
c) What is the marginal cost per ton when
1,000 tons have been extracted?
8. Let f (t) be the number of centimeters of
rainfall since midnight. t is the time since
midnight, in hours.
Approximate answers are okay.
(a) When did the skydiver’s parachute open?
(b) When did the skydiver land on the ground?
(c) During freefall, an object reaches a terminal
velocity where the object no longer gains
speed due to air resistance. What is the
skydiver’s terminal velocity during his
freefall?
6. Find the values of A and B that make this
function both continuous and differentiable.
ì Ax 3 + B x £ 4
y =í
î 4 x + 12 x > 4
a) What are the units of f ¢( t) ?
Explain the practical meaning of the
following statements:
b) f (7) =1.5.
c) f ¢(7) = .15
d) f ¢(9) = 0
9. The following table gives the position of a
moving body as a function of time. The
position is measured in feet from some fixed
point. The time is in seconds.
t
0 0.5 1 1.5 2 2.5 3 3.5 4
(s)
p
10 38 58 70 74 70 59 38 10
(ft)
a) What is the body’s average velocity
between t = 0 and t=4?
b) What is the average velocity between t =
2.5 and t=3.5?
c) Estimate the instantaneous velocity of the
body at t=2.75?
d) Approximately when does the body
change direction?
Chapter 3: Derivatives – Test Review
Calculus
page 12
Answers to: Chapter 3: Derivatives – Test Review, Part 2
1. a. i. nowhere, ii. x = 2, cusp
c. i. x = 2, jump; ii. x = 2, discontinuity
2. a.
3. a. about t = 3, 5
c.
b. i. nowhere, ii. x = 4, corner
d. i. nowhere, ii. x = -1, vertical tangent
b.
b. between t = 3 and 5
c.
5. a. about t = 60 second
b. about t = 104 seconds
c. about 185 mph
6. A = 1/12, B = 68/3
7. a. $ / ton of ore
b. 200 $ / ton
c. 250 $ / ton
4. a. about t = 9.2
b. t = 7
c. greatest acceleration between t = 3
and 4, greatest acceleration is between
t = 4 to 7
d.
8. a. cm/hr
b. after 7 hours (at 7am) it has rained 1.5 cm
c. at 7am it is raining at .15 cm / hour
d. at 9 am it has stopped raining
9. a. 0 ft/sec
c. -22 ft/sec
b. -32 ft/sec
d. about t = 2 sec