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AP Calculus Assignments: Derivative Techniques
Day
Topic
Assignment
1
Basic Differentiation Formulas
HW Derivative Techniques - 1
2
Product and quotient rules
HW Derivative Techniques - 2
3
Derivatives of exponential and log functions
HW Derivative Techniques - 3
4
Derivatives of Trig Functions
HW Derivative Techniques - 4
5
Rates of Change
HW Derivative Techniques - 5
6
Practice
7
The Chain Rule
HW Derivative Techniques - 7
8
The Chain Rule
HW Derivative Techniques - 8
9
Implicit Differentiation
HW Derivative Techniques - 9
10
More practice
HW Derivative Techniques - 10
11
Derivatives of inverse functions
HW Derivative Techniques - 11
12
Still more practice
HW Derivative Techniques - 12
13
Differentiability
HW Derivative Techniques - 13
14
Review
HW Derivative Techniques - Rev
15
**TEST**
**QUIZ**
**QUIZ**
**QUIZ**
HW Derivative Techniques - 6
AP Calculus HW: Derivative Techniques - 1
Differentiate the following:
5. g  x    5 x 
3
6. y  3t 2 
4
4. V  r    r 3
3
3. f  v   v2  8v  20
2. y  5t 2/5
1. y  3x5
2
t2
7. f  u  
8. Find the equation of the line tangent to y  x 
u 2  3u  4
u
8
at the point where x = 2.
x
9. Find the equation of the line tangent to y  x 2  x at the point where x = 1.
10. A bug travels along the x-axis. Its position as a function of time is given by x  t 3  3t (x in meters, t in
seconds; it’s a very fast bug) for t  0.
a. Find the bug’s velocity as a function of time.
b. Find the bug’s acceleration as a function of time.
c. Find the bug’s acceleration when its velocity is 0.
11. Solve f '  x   g '  x 
f  x
 g  x  q '  x  for q'(x) and prettify it (no complex fractions). (Note: this is an
g  x
algebra problem; not a calculus problem. Do it anyway. And mind your gs and qs.)
12. Kenny had to take the derivative of y  13 . He wrote y '  1 2 . What happened?
3x
x
AP Calculus HW: Derivative Techniques - 2
Differentiate using the product and quotient rules.
1  x2
1. f  x  
2. v   t 2  t  1 t 2  2 
2
1 x


3. f  u   1  u  u  u 3 
Differentiate using the most appropriate method. In #6  8, a and b are constants.
4. y  3x
2
x4
5. f  x  
2
12
6. F  r    2
r
9. Find the equation of the tangent line to the graph of y 
7. y  ax  b x
5
1
at the point where x = –2.
1  x2
10. Suppose f  4  5 , f '  4   0.5 , g  4  4 and g '  4  2 . Evaluate
a.
 fg  '  4
 f '
b.    4 
g
t3 b

8. g  t  
a
t
g'
c.    4 
f 
(This assignment continued on next page.)
11. If h  2   5 and h '  2  2 , evaluate
d  h  x 
b.


dx  x  x 2
d
a.
 xh  x  
dx
x2
Notation: the vertical bar with x = 2 at the bottom
means "evaluated at x = 2." The evaluation must be
done after the derivative is taken. Why?
12. The graphs of f and g are shown at right. Use them to evaluate
 f '
a.  fg  ' 1
b.    5
g
y
f
g
13. If h is a differentiable function, find an expression for each of the
following:
h  x
1  xh  x 
x2
a. y  x 2 h  x  b. y  2
c. y 
d. y 
h  x
x
x
x
14. a. Use nDeriv on your calculator to graph f  x   e x and its derivative. (Enter Y = nDeriv(e^(X),X,X))
What is the derivative of f  x   e x ? If the graph doesn’t do it for you, check the tables.
b. Use nDeriv to make a table of values for the derivative of f  x   ln x for x = 1, 2, 3, 4, 5, 6, . . . From
the table, guess the derivative of f  x   ln x .
eh  1
.
h 0
h
15. Use your calculator to estimate the value of lim
3
2
16. Kenny had to take the derivative of y  x 2 4 x . He wrote y '  3x  4 . What happened?
2x
x 1
AP Calculus HW: Derivative Techniques - 3
Differentiate the following:
1. y  2e
x
ex
2. f  x  
3
3. y  6 ln x
5. y  2 x 2  3e x  4
6. f  t   4 t  2ln t  1
8. f  r   r 2er
9. g  u   eu u
11. y 
ex
1 x
12. s   t 2  2t  et
4. g  u  
ln u
4
7. y  5 x 
ln x
3
ez
z2
ax  b
13. y 
cx  d
10 h  z  
14. If f  x   e x g  x  where g(0) = 5 and g'(0) = –2, find f '(0).
(This assignment continued on next page.)
15. Review the graphs of y  sin x and y  cos x and then use nDeriv on your calculator to graph the
derivatives of those functions. From the graphs, determine the derivatives of y  sin x and y  cos x .
16. A problem on a long-ago homework assignment said: Estimate the value of the following limits. Then put
the answers in your brain.
sin x
1  cos x
a. lim
b. lim
x 0
x

0
x
x
Retrieve those answers from your brain. If you can’t find them, redo the problem.
17. Kenny did problem #16 with his calculator in degree mode. What happened?
AP Calculus HW: Derivative Techniques - 4
1. Memorize the derivative formulas for the six basic trig functions.
Differentiate:
2. y = sinx + cosx
5. y 
tan t
t
3. y  x 2 cos x
6. y 
sin x
x  cos x
4. f  z   2cot z  z sec z
7. y  r   er  tan r  r 
8. Find the equation of the tangent line to the graph of y  tan x at the point where x = /4.
9. Find the equation of the tangent line to the graph of y  x cos x  x at the point where x = .
10. If r   2 cos  , find expressions for r'() and r"().
d 99
11. Evaluate 99  sin x  . (Try to do this without actually taking 99 derivatives!)
dx
x
(g is called the signum function, sgn(x)).
x
b. What is the relationship between the functions f and g above?
12. a. Draw the graphs of f  x   x and g  x  
13. Without looking at your notes, write down the derivative formulas for the six basic trig functions. If you
can’t do this, go back to problem #1.
14. Kenny thought he could survive AP Calc without knowing his derivative formulas cold. What happened?
AP Calculus HW: Derivative Techniques - 5
1. A bug moves along the x-axis with position x  t 3  12t 2  36t for t  0 (x in meters, t in seconds.)
a. Find the bug’s velocity at time t.
b. When is the bug at rest?
c. When is the bug moving forward?
d. Find the bug’s acceleration as a function of time.
2. a. Find the rate of change of the circumference of a circle with respect to its radius.
b. Find the rate of change of the area of a circle with respect to its radius.
3. In a right triangle, one acute angle measures  and the leg adjacent to that angle measures a.
a. Express the length of the opposite leg in terms of a and .
b. Find the rate of change of the opposite leg with respect to a (assuming  remains constant) and
evaluate for  = 0.5 and a = 100 cm. What are the units?
c. Find the rate of change of the opposite leg with respect to  (assuming a remains constant) and
evaluate for  = 0.5 and a = 100 cm. What are the units?
4. A tank holds 4800 gallons of water that drains through a hole in the bottom in 40 minutes. Torricelli’s Law
2
t 

says the volume of water in the tank after t minutes is given by V  t   4800 1   .
 40 
a. Find the rate at which water is draining from the tank after 10 minutes.
b. What does the sign of your answer in part a indicate?
c. When is the water flowing out of the tank the fastest? Slowest?
5. a. Find the instantaneous rate of change of the volume of a cube as a function of its edge length x, in cm.
b. Find the average rate of change of the volume of a cube on the interval 2 cm  x  11 cm.
c. For what value of x does the instantaneous rate of change of the volume equal the average rate of change
on the interval [2, 11]?
6. Boyle’s Law says that for an “ideal gas” at a constant temperature, PV = k (pressure in pascals times
volume in liters equals a constant).
a. For what value of pressure will the instantaneous rate of change of volume with respect to pressure be
the same as the average rate of change of volume on the interval 2 Pa  P  8 Pa?
b. A sample of gas at low pressure is steadily compressed (pressure is increased) at a constant temperature
for a period of several minutes. Is the volume changing more rapidly at the beginning or the
end of the period? Justify your answer.
7. Newton’s Law of Gravitation says that two masses M and m attract each other with a force of
GMm
dF
F  2 where G is a constant. Find
. Why is it negative?
r
dr
8. Review the two limit definitions of the derivative f '(a).
9. Kenny was given the function y = xlnx for x  [1, e] and asked to find the rate of change of y with respect to
y  e   y 1
 e . What happened?
x. He did
e 1
e 1
AP Calculus HW: Derivative Techniques - 6
1. The functions f and g are differentiable for 1  x  4. The values of the functions and their derivatives at
selected points are given in the table below.
a. Evaluate 3 f  2 g  1 .
'
b. Evaluate  fg  '  3 .
'
x
1
2
3
4
f(x)
6
9
10
1
f '(x)
4
2
4
3
 f 
c. Evaluate    3 .
g
d. Write an equation for the line tangent to the graph of g at x = 1.
e. Estimate g"(2.)
g(x)
2
3
4
6
g'(x)
5
4
2
1
y
f
2. The graphs of f and g are shown at right. Use them to evaluate
d  g  x 
a.  fg  '  5
b.


dx  f  x  
x 5
g
x
3. Differentiate the following:
c. g  w   3 w 
e. y  3s 2  4s  7
1
ln t
2
f. y = ex
i. y = e3
j. f  t   t 2 cos t
k. f   
a. y = 3sin x
b. y 
2
w
d.
f u  
6
u3
h. y = sec
g. y = 3x
tan 

l. f x  
x
3

1
x
2
4. The mass m in grams of a spherical snowball is a function of its radius r in centimeters: m = f(r).
a. Using correct units, tell what f(4) = 268 means.
b. Using correct units, tell what f '(4) = 201 means. Give an alternate notation for f ' in this context.
c. Using the values from (a) and (b) above, estimate the mass of the snowball when
(1) r = 4.1 cm.
(2) r = 3.8 cm.
d. What is the appropriate calculus notation for the rate of change of the mass of the snowball with respect
to time t ?
5. Let f be a differentiable function of with f(5) = 2 and f '(5) = 3. Find the equation of the line tangent to the
graph of g(x) = x2f(x) at x = 5.
6. Let f be a differentiable function of with f(9) = –4 and f '(9) = 2. Find the equation of the line tangent to the
f x 
graph of hx  
at x = 9.
x
7. a.
b.
c.
d.
e.
Sketch a graph of a continuous function f having f ' > 0 for all x.
Does the function you sketched in part (a) have an inverse function? HDYK?
Hypothesize a general rule about the derivative of a continuous function f and the invertibilty of f.
Use calculus to prove that the function f(x) = x3 + 4x – 5 has an inverse function.
Use calculus to prove that g(x) = x3 – 4x – 5 is not invertible (unless we restrict the domain).
8. Kenny had to take the derivative of y  sin2 x . He wrote y '  cos x . What happened?
2x
x
AP Calculus HW: Derivative Techniques - 7
Differentiate.
4
1. y   5 x  1
5. g  r  
4
1  3r 
2
1 
9. y  6 tan  x 
2 
13. y = 4cos3(x)
2. a  t   3e2t
3. f    2sin  3 
4. y  ln  e x  1
6. y  x 2  4
7. y   ln  cos 
8. y   x 2  3x  5
10. y  4e
14. g  r  
t
1
3
4r  1
11. f  u   u 2  2u  5
12. y  sec  5 x 2 
15. f  z   ze  z
16. y 
2
sin
2
 x
ex  1
2
17. If F  x   f  g  x   , f(2) = 4, f '(2) = 1.5, f '(–5) = 6, g(2) = –5 and g'(2) = 0.5, evaluate F '(2).
18. The functions f and g and are shown in the graph at right. Use the graph to
evaluate the following:
a.
y
f
'
 fg  ' 1
c.  f  g  x    ' 1
e.  g  g  x    ' 1
f 
b.   1
g
d.  g  f  x    ' 1
g
x
AP Calculus HW: Derivative Techniques - 8
Differentiate:
1. f  x   e
 tan x
1 
5. y  12 tan 2  x 
4 

9. g  x   ln 1  x

t
6
2. y  ln  x  1
3. f  t   e
6. f    1  sin 2 
7. y  ln  sec  tan  
2
10. G  x   ln x
11. f  t  

1
4
t4 1
4. f  z   0.25e z
2
8. f  t   3cos  e 2t 
 ar 
12. f  r   ln 

ar 
In #13 – 16, f is a differentiable function.
13. y  f  x 3 
15. y  f  sin 2 x 
14. y   f  x  
3
f ln x
16. y  e  
17. Selected values of a differentiable function f are shown in the table at right. Use
them to evaluate
d
d
a.
b.
xf x
f f  x2 
dx
dx
x 9
x 3
  
 

x
3
6
9
f(x)
9
7
6
f '(x)
-4
-2
0.25
2
AP Calculus HW: Derivative Techniques - 9
Use implicit differentiation to find
1. y 3  y  3x
dy
for the following:
dx
2. x 2  xy  y3  10
dy
for the function ln  x  y   y .
dx
a. Jess applied implicit differentiation to the problem in its original form. What did she get?
b. Chris used exponentiation base e to eliminate the natural log, getting x  y  e y , and then differentiated
implicitly. What was his answer?
b. Jess and Chris then spent 15 minutes arguing about who was right. Who was? Justify your answer.
3. Chris and Jess had to find
4. Find the slope of the graph of  y  3 exy  x3 11 at the point where y = 0.


1
5. Find the slope of the graph of y  tan   x 2  y 2  at the point  ,
2

1 .

6. Find an equation of the tangent line to the graph of y 4  3x 2 y  4 x  125 at the point in the fourth quadrant
where x = 2.
7. The graph of x 2  6 xy  25 y 2  16 is an ellipse. Find the coordinates of the points where the ellipse has
horizontal tangent lines.
AP Calculus HW: Derivative Techniques - 10
Yes, it’s a lot of problems. If you are getting good at this, it shouldn’t take so long. If you are not yet good at
this, you need the practice. Start differentiating.
Differentiate the following.
1
1. y  sin  6 x 
2
2. ƒ(x) = aebx
3. y = 2tan(4x – 1)
4. y  6 3 x 
5. y = e2
6. y  cos  x 2 
7. y  cos 2 x
8. y   x 2  2 x  5
9. f  x   3e x
13. y 
2
1 x
1 x
14. f  x  
2
17. Use the table of values for f, f ', g and g' to evaluate h'(2) for
a. h(x) = x2 – 4g(x)
b. h(x) = f(x)g(x)
2
c. h  x    g ( x)
d. h(x) = xe–g(x)
f. h  x  
e. h(x) = f(g(x))
g. h  x  
f  x
x3
h. h  x   f  x 2 
x
g  x
x2  2
4x
x
0
1
2
3
4
f
2
1
3/2
5/2
4
f'
-2
0
-1/2
3/2
2
g
4
4
3
1/2
1
Find y' for each of the following:
18. y3  xy 2  4
19. x  cos  x  y   0
AP Calculus HW: Derivative Techniques - 11
1. If f is a one-to-one function, f(3) = 5 and f '(3) = 2, which of the following is true?
1
1
(A) (f –1)'(3) =
(B) (f –1)'(5) =
(C) (f –1)'(5) = 2
2
2
1
1
(D)(f –1)'(2) =
(E) (f –1(5))' =
5
2
2. If f(x) = x3 + x – 7, find (f –1)'(3).
3. If f  x   x 2 
4
, x > 0, and g(x) = f –1(x), find g'(15).
x
4. If x = y3 +2y + 4, find
4
sec 2 x
15. f    ln 1  sin 2  3   16. y  1  e  
1
1 x
12. f  x  
11. ƒ(z) = z2e–3z
10. ƒ(r) = 2πr(r + h)
3
x2
dy
.
dx x 7
(This assignment continued on next page.)
g'
1
-1/2
-2
-1
1/2
5. The function f is one-to-one and differentiable. Selected values
x
f(x)
  3 .
of f are shown in the table. Estimate the value of f
1 '
2.9
3.3
6. The function f is graphed at right. The graph consists of two line segments.
d 1
Find
without graphing f –1.
f x 
dx
x 3
3.0
3.2
3.1
3.0
3.2
2.7
y
f
7. Use the method we used in class to find the derivative of sin–1x to find
derivatives for
a. tan–1x (memorize the answer) and
b. ln x (I know you already know the answer; practice the technique.)
x
AP Calculus HW: Derivative Techniques - 12
Differentiate the following:
2. f  x   2e3x
1. y = 3sin (4x)
5. f  x  
4
x
4. y 
x2  2
x
x 3
10. f(x) = e–x cos (2x) 11. y  2
x  3x
3
3
3. y 
6. f(x) = tan (2x)
x2
9. f(x) = x2ln x
14. y  sec  x 3 
13. f(x) = 6x2e–0.5x
7. y 
1 4 4 3
x  x  3x 2  5 x  1
2
3
8. y = tan–1 x
12. f  x    x 2  2 x  4 
3
15. y = sin–1 x
16. f(x) = ln(x2 – 3x + 4)
y
17. Use the graph at right to evaluate the following:
a.
 fg  ' 3
d.
d
dx
 g  x 
'
 f 
b.    5 
g
2

e.
x 0
d
 xf  x 
dx
x 3
Find y' for each of the following:
18. 3x 2  2 xy  5 y 2  1
19. sin  xy   x
d
c.
f  g  x 
dx
x 1
f.
d  x2 


dx  g  x  
f
g
x 3
x
AP Calculus HW: Derivative Techniques - 13
2
 x
1. Determine if f  x   
 x
x 1
x 1
is differentiable at x = 1.
3x 2
x 1
2. Find the values of a and b so that f  x   
is differentiable at x = 1.
ax  b x  1
 x2
3. Let f  x    3
x
x0
x0
. Find ƒ'(0) and ƒ"(0).
4. Differentiate the following:
a. y  1  3 2 x
b. f  x   4sin 2x  3cos 2x
d. y  sin 1  4 x 
1
g. y  x tan  
 x
3
c. y  1  4 tan 2  2 x 
1
e. f  x   cos 2  3e 2 x 
4
f. y  3 1  e x / 3
h. f  x   ln  x  sin  2x  
tan 1  3 x 
i. y 
1  9x2
5. Find the equation of the tangent line to
x
2
 y 2   50 xy  0 at the point (2, 4)
2
AP Calculus Review: Derivative Techniques
1. Differentiate the following.
a. y  sin x
b. y  sin 2 x
e. f  x    0.5 x 2  cos 2 0.5 x 
c. f  x  
x
f. f  x   sin 1  
2
4
x
d. y  2 cos3
x 1
2
g. g  x  
ln  tan x 
tan x
4
2. Find the slope of the graph of y  4 x 3 at the point (8, 64).
3. Find an equation of the tangent line to y  2 x 
4
at the point where x = 4.
x
1
4. If ƒ(1) = 3 and ƒ'(1) = –2 , find an equation of the tangent line to y = ƒ(x) where x = 1.
(This assignment continued on next page.)
1
x
2
5. Suppose f and g are differentiable functions.
Selected values of f and g and their first derivatives
f ' and g' are shown in the table. Evaluate the
following:
a.
d
 f  x  g  x 
dx
x2
b.
d
g  f  x 
dx
x 0
c.
3
d
g  x 

dx
x 2
f.
e.
6. If f  x  
x
f(x)
f '(x)
g(x)
g '(x)
d  f  2 x  


dx  2 x 2  x 1
2
4
1
7
1
d.
1
6
1
5
1.5
0
2
2
3
2

d x2  4
e g  2x 
dx
1
2
1
4
2
3
0
1
0.5 1.5

3
2
2.5
2
1
x2
d 1
g  x
dx
x 3
f  2  h   f  2
2 6
x  4 x 3  1 , evaluate lim
.
h 0
3
h
7. Find the equation of the tangent line to the graph of x ln y  2e  4 y at the point where y = e2.
8. The itsy bitsy spider is climbing up the water spout. Her position in centimeters as a function of time in
seconds is given by y(t) = 150tan–1(t), t  0.
a. Find Itsy’s velocity as a function of time.
b. Find Itsy’s acceleration as a function of time.
c. Is Itsy’s speed increasing or decreasing? Justify your answer.
9. Find the equation of the tangent line to the graph of x3 – 3xy + y2 +19 = 0 at the point in the second
quadrant where x = –4.
10. If f(x) = 2x5 + x3 + 1, find (f –1)'(–2).
 tan 1 x

11. Let f  x    a
 b
x
x 1
x 1
. For what values of a and b will f  x  be differentiable at x = 3?
12. A the volume of a right circular cone with a slant height of l, in meters, and a
1
base angle of  is given by V   l 3  sin   sin 3   .
3
a. Find the rate of change of the volume with respect to the slant height if the
base angle is held constant.
b. Find the rate of change of the volume with respect to the base angle if the
slant height is held constant.
c. What are the units of the answer in part b above?
l

13. Find the points where the graph of 2 y 3  x 2  4 x  6 y  1  0 has horizontal or vertical tangents.
4
0
3
4
2
Answers to selected problems:
HW – 1
1. y '  15 x 4
5. g '  x   375x2
6. y '  6t 
8. y  6  ( x  2)
9. y  2 
4
t3
HW – 2
7.
10a.
12a.
13a.
3
3
7. f '  u   u1/ 2  u 1/ 2  2u 3/ 2
2
2
b. a  t   v '  t   6t
c. 6 m/s2
g  x f ' x  f  x g ' x
 g  x  
2
1  x   2x   1  x  2x 

2
2
4 x
2. V '   t 2  t  1  2t    t 2  2   2t  1
2 2
(1  x )
(1  x )
24
1
4. y '  6 x
5. f '  x   2 x3
6. F '  r   3
f '  u   1  u 1  3u 2    u  u 3 
r
2 u
1/2
1 4
3t
b
b
8. g '  t  
9. y    x  1
 3/2
y '  5ax 4 
2 25
2a 2t
2 x
8
b. –3 c. 12/25
11a. 1
b. –9/4
0
b. –2/3
x 2 h '  x   2 xh  x 
2 xh  x   x 2 h '  x 
2
c. y ' 
y '  2 xh  x   x h '  x  b. y ' 
2
x4
 h  x 
1. f '  x 
3.
4. V '  r   4 r 2
5
( x  1)
2
10a. v  t   x '  t   3t 2  3
11. q '  x  
3. f '  v   2v  8
2. y '  2t 7/5
2 2


x  h  x   xh '  x   
d. y ' 
1
2 x
1  xh  x  
x
HW – 3
1. y '  2e
x
5. y '  4 x  3e x
9. g '  u  
1
2 u
ex
2. f '  x  
3
2 2
6. f '  t  

t t
eu  eu u
10. h '  z  
12. s '   t 2  2t  et   2t  2  et   t 2  2  et
14. 3
3. y 
6
x
7. y '  5 
1
3x
1
4u
8. f '  r   r 2er  2rer
1  x  e x  e x  xe x
2
2
1  x 
1  x 
a  cx  d   c  ax  b  ad  bc
y' 

2
2
 cx  d 
 cx  d 
z 2e z  2 ze z
z4
13.
4. g  u  
11. y ' 
HW – 4
2. y '  cos x  sin x
5. y ' 
3. y '  2 x cos x  x 2 sin x
t sec2 t  tan t
t2
6. y ' 
1  sin x  x cos x
 x  cos x 
2
4. f '  z   2csc2 z  z sec z tan z 
1
2 z
sec z
7. y '  r   e r  tan r  r   e r  sec2 r  1


8. y  1  2  x  
9. y  2  2  x   
4

10. r '    2 cos   2 sin  ; r "   2cos  4 sin    2 cos
11. –cos x
d
x
14. Kenny failed the course miserably. But because his school foolishly paid for
x
dx
x
the AP exam, Kenny took it anyway. He got a 1. Then he died horribly.
12b. g = f ', or
HW – 5
1a. v(t) = 3t2 – 24t + 36
b. t = 2 and t = 6
c. [0, 2)  (6, )
d. a(t) = 6t – 24
dC
dA
dA
2a.
= 2
b.
= 2r (Note:
= C)
dr
dr
dr
dL
dL
 129.845 cm/rad
3a. L  a tan 
b.
 0.546 cm/cm c.
d a 100, 0.5
da  0.5
4a. –180 gal/min
b. water flowing out of the tank
c. at t = 0; at t = 40
dV

V
 3 x 2 cm3/cm
 147 cm3/cm
5a.
b.
c. 7 cm
dx
x
dV
6a. 4 Pa
b. At the beginning since as P increases,
will decrease.
dP
dF
2GmM

7.
; as r increases, F decreases.
dr
r3
HW – 6
1a. 2
3a. 3cos x
b. 4
1
b.
2t
h. sec  tan 
c. –9/4
d. y – 2 = 5(x – 5)
1 2/3
c. w  w3/2
d. –18u–4
3
i. 0
j. 2tcos t –
t2sin
t
e. –3/2
2a. 1/6
b. 7/24
e. 6s – 4
f. ex
g. 3xln3
 sec2   tan 
k.
2
l. 5x4 + 4x – x–2
4a. When the radius is 4 cm, the mass is 268 g.
b. When the radius is 4 cm, the rate of change of the mass with respect to radius is 201 g/cm.
c1) 288.1 g
2) 227.8 g
4 15
5. y – 50 = 95(x – 5)
6. y    x  9 
3 16
7b. Yes; f is one-to-one by the HLT.
c. If f ' is always positive or always negative, f will have an inverse function.
d. f '(x) = 3x2 + 4 which is positive for all x so f has an inverse.
e. g'(x) = 3x2 – 4 which is neither always positive nor always negative so f is not invertible.
8. Kenny died horribly. Because it was the second time he had made the same really annoying mistake, his
irate math teacher had his body blasted into its individual atoms which were in turn reduced by fission to so
many hydrogen atoms. The energy thus released was used to disperse those atoms to the far reaches of the
universe. Kenny was never heard from again.
HW – 7
1. y '  20  5 x  1
2. a '  t   6e2t
3
24
5. g '  r  
1  3r 
x
6. y ' 
3
1 
9. y '  3sec 2  x 
2 
10. y ' 
x 4
2
2
e
t
4
13. y '  12cos 2 x sin x 14. g '  r  
16. y ' 
e
x2
3 3  4r  1
  x   2 1 x   sin  x  e
 e  1
 1 cos
7. y' = tan
8. y '  2  x 2  3 x  5   2 x  3
x2
4
u 1
u  2u  5
2
15. f '  z   1  2 z 2  e z
b. 4/3
12. y '  10 x sec  5 x 2  tan  5 x 2 
2
 2x 
17. 3
2
x2
18a. 0
4. y ' 
11. f '  u  
t
ex
ex 1
3. f '    6cos  3 
c. 3/4
e. –2
d. undefined
HW – 8
1. f '  x   e tan x sec2 x
2. y ' 
2x
2
x 1
sin  cos 
 x
 x
5. y '  6 tan   sec 2   6. f '   
4
4
1  sin 2 
1
1
9. g '  x  
10. G '  x  
2 x ln x
2 x 1 x


1 t
3. f '  t    e 6
6
4. f '  z   0.5 ze z
7. f '    sec
8. f '  t   6e 2t sin  e 2t 
11. f '  t  
14. y '  3  f  x   f '  x 
15.
e f  ln x  f '  ln x 
16. y ' 
x
17a. 3
b. –3
HW – 9
1. y ' 
3
2. y ' 
3y 1
12
4. y '  
5
2
5.
2
4  1
7. Horizontal tangents at  3,
HW – 10
1. y' = 3cos 6x
5. y' = 0
9. f '  x   6 xe x
2
2x  y
x  3y2
3a. y ' 
6. y  3 
1
x  y 1
5/ 4
4
2a
a  r2
2
2
b. y ' 
1
1 ey
1
 x  2
3

1 (and vertical tangents at  5,

2. ƒ '(x) = abebx
3. y' = 8sec2(4x – 1)
6. y' = –2xsin x2
7. y' = –2cos x sin x
10. ƒ '(r) = 4πr + 2h
12. f '  r  
t  1
y '  2 f '  sin x  sin x cos x
13. y '  3 x 2 f  x 3 
2
t 3
2
3
 if you want a little more practice)
5
11. ƒ'(z) = 2ze–3z – 3z2e–3z
4. y' = 2x–2/3 + 6x–3
8. y '  4  x 2  2 x  5  2 x  2 
3
12. f '  x  
1
1
 2
4 2x
2
13. y ' 
1  x 
e
16. y ' 
sec 2 x 
x
15. f '   
1  x 
2 3
6sin  3  cos  3 
1  sin 2  2 
sec  2 x  tan  2 x 
1 e 
b. –4.5
sec 2 x 
17a. 12
18. y ' 
14. f '  x  
2
y
3y  2x
c. –12
d. 5e–3
e. –3
3. 4/33
3. y '  
2. f '(x) = –6e–3x
5. f '  x   2 x5/ 3
6. f '(x) = 2sec2 (2x) 7. y ' 
12. f '  x   3  x 2  2 x  4   2 x  2 
2x  3
( x  3x  4)
1 x
17a. 4
b. –9/8
c. 2
sec  xy   y
y  3x
18. y ' 
19. y ' 
x
5y  x
16. f '  x  
2
2. a = 6, b = –3
2
3
4x
3 1/ 2
x  x 3/ 2
2
d. –16
1  2 cos  2 x 
x  sin  2 x 
e. 19/2
1
cos x
f. 3/2
i. y ' 
c. y '  16 tan  2x  sec2  2 x 
3  18 x tan 1  3 x 
5. y  4  
1  9 x 
2 2
b. y ' 
2 x
1  1 
d. y '  3cos 2  x  sin  x 
2  2 
1
f. f '  x  
4  x2
cos 2 x
sin 2 x
c.
f ' x 
d. y ' 
2
 x  2
11
1
( x  1)3/ 2
2
e. f '  x   4  0.5 x 2  cos 2 0.5 x   x  cos  0.5 x  sin  0.5 x  
3
g. g '  x  
1  ln  tan x 
sin 2 x
4
1  16 x 2
1
1
g. y '  3 x 2 tan    x sec 2  
x
 x
2 / 3
1
f. y '  e  x / 3 1  e x / 3 
9
Review
1a. y' = y ' 
1
1  x2
 x2  6 x  9
11. y ' 
( x 2  3 x) 2
8. y ' 
3. ƒ'(0) = 0, ƒ"(0) DNE
b. f '  x   8cos 2 x  6sin 2 x
2
4. y' = 2x3 – 4x2 + 6x – 5
2
e. f '  x   3e2 x cos  3e2 x  sin  3e2 x 
h. f '  x  
4
x2
13. f '  x   3x  4  x  e0.5 x 14. y '  3 x 2 sec  x3  tan  x3 
2
1
6. –2
10. f '  x   e x  2sin  2 x   cos  2 x  
9. f '(x) = x(1 + 2ln x)
4a. y ' 
h. 16
5. –2/5
4. 1/5
1. y' = 12cos (4x)
HW – 13
1. No.
g. 7/9
19. y '  csc  x  y  1
HW – 11
1. (B)
2. 1/13
d
1
tan 1 x 
7a.
dx
1  x2
HW – 12
15. y ' 
f. –11/32
5a. 4
b. 3
2
7. y  e  2e  x  3e 
8a. v  t  
1
( x  4)
4
c. 5
3. y  5 
2. 32/3
150
1 t2
b. a  t  
300t
1  t 
2 2
1
4. y  3   ( x  1)
2
d. 20
e. 147
f. 1/2
6. 80
c. Speed is decreasing b/c v and t have opposite signs when t > 0.
1
 2
11. a   , b 
2
4
dV
dV 1 3
  l 2  sin   sin 3  
  l  cos   3sin 2  cos  
12a.
b.
c. m3/radian
dl
d 3
13. Horizontal tangent at (2, 2.054); vertical tangents at (1, 1), (1, 1), (3, 1) and (5, 1).
9. 13/6
10. 1/13