Homework




Homework Assignment #16
Read Section 3.8
Page 178, Exercises: 1 – 93 (EOO)
Quiz next time
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 178
Fill in the table.
1. f  u   u
3
2
g  x   x4  1
f  g  x 
 x  1
4
3
2
f  u 
f  g  x 
3 12
u
2
1
3 4
2
x

1
 
2
g x 
4x
3
g 
f
6 x  x  1
3
4
1
2
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 178
Write the function as a composition and find the derivative.
5. y   x  sin x 
4
y   x  sin x   f  u   u 4  g  x   x  sin x  y  f  g  x  
4
f   4u 3  g   1  cos x  y   4  x  sin x  1  cos x 
3
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 178
Find the derivative.
9. y   x  9 
2
4
y   x  9  f u   u 4  g  x   x2  9  y  f  g  x 
4
2
f   4u  g   2 x  y   8 x  x  9 
3
2
3
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 178
Find the derivative.
10  x 2
13. y  e
10  x 2
ye
 f  u   eu  g  x   10  x 2  y  f  g  x  
f   e  g   2 x  y   2 xe
u
10  x 2
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 178
Find the derivative of f ○ g.
u
g  x   x  x 1
17. f  u   e
f  u   eu , g  x   x  x 1  f   u   eu , g   x   1  x 2
2
x  x 1

 f g   1  x  e
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 178
Find the derivative.
21. y  sin x 2
y  sin x 2  sin u  u  x 2  u   2 x
y  cos x 2  2 x   y  2 x cos x 2
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 178
Find the derivative.5
2
2
y

t

3
t

1


25.
y   t  3t  1
2
5
2
u
5
2
 u  t 2  3t  1, u   2t  3
7
5 2
y    t  3t  1 2  2t  3
2
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 178
Find the derivative.
29. y  cos3 e4
y  cos3 e 4  u 3  u  cos e 4  cos v  v  e 4
u   sin v v  v  4e 4
y  3u 2u   sin v v   3cos 2 e 4   sin e 4  4e 4 
y  12e 4 sin e 4 cos 2 e 4
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 178
Find the derivative.
1
33. y  e x
ye
1
x
 eu  u  x 1 , u    x 2
1
x
e
y  e x   x   y   2
x
1
2
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 178
Find the derivative.
37. y  x cos 1  3 x 
y  x cos 1  3 x   u  1  3 x, u   3
y  x   sin  u  u    cos u 1  3 x sin 1  3 x   cos 1  3 x 
y  3x sin 1  3 x   cos 1  3 x 
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 178
Find the derivative.
4
3
41. y   x  cos x 
y   x  cos x   u  x 3  cos x, u   3x 2  sin x
4
3
y  4  x  cos x 
3
5
 3x
2
 sin x 
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 178
Find the derivative.
4
3
45. y   z  1  2 z  1
y   z  1  2 z  1  u 4 v 3  u  z  1, u   1  v  2 z  1, v  2
4
3
y  u 4  3v 2 v   v 3  4u 3 u  

  z  1 3  2 z  1
4
2
 2     2 z  1
3
 4  z  1 1
3
y  6  z  1  2 z  1  4  2 z  1  z  1
4
2
3
3
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 178
Find the derivative.
1
2
2
y

cos
6
x

sin
x


49.
y   cos 6 x  sin x
2

1
2
  cos u  sin v 
1
2
w
1
2
u  6 x, u   6  v  x 2 , v  2 x  w  cos u  sin v
w   sin u  cos v
1
1
2  2
y   cos 6 x  sin x    sin u  6   cos v  2 x  
2
1
1
2  2
y   cos 6 x  sin x   2 x cos x 2  6sin 6 x 
2
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 178
Find the derivative.
1
1
z 1  z 1  2
2


u
53. y 

z 1  z 1 
z  11   z  11

z 1
2
u
, u 

2
2
z 1
 z  1
 z  1
1  z 1 
y  

2  z 1 
y    z  1
1
2
2
1
 2 
2
2  z  1 

    z  1 

2
  z  1 
z

1




 z 1 


z

1


1
2
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 178
Find the derivative.
57. y  cot 7  x5 
y  cot 7  x 5   u 7  u  cot  x 5   cot v  u    csc 2 v v
v  x 5  v  5 x 4


y  7u 6 u v  7 cot 6  x 5   csc 2  x 5   5 x 4 
y  35 x 4 cot 6  x5  csc 2  x 5 
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 178
Find the derivative.
x
2 x
y

4
e

7
e
61.
y  4e  x  7e 2 x  y  4   1 e  x   7   2  e 2 x 
y  4e  x  14e 2 x
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 178
Find the derivative.
2 t
y

cos
te


65.
y  cos  te 2t   cos w  w  u v  u  t , u   1; v  e 2t , v  2e 2t
w  uv  vu  2te 2t  e 2t 1  e 2t 1  2t 
y   sin w w   sin  te 2t   e 2t 1  2t  
y  e 2t  2t  1 sin  te 2t 
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 178
Find the derivative.
69. y  1  1  x
y  1 1 x  u  u  1 1 x  1 v  v  1 x
v 
1
1
 u 
v 
2 x
2 v
1
1
1
1


y 
u 
2 u
2 1 1 x 2 1 x 2 x
y 
1
8 x 1 x 1 1 x
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 178
73. Compute
df
df
du
if
 2 and
 6.
dx
du
dx
df df du

 2 6  12
dx du dx
df
 12
dx
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 178
77. Compute the derivative of h  sin x  at x 
h  0.5   10.

6
, assuming that
d
 

3

h  sin x   h  sin x  cos x   h  sin  cos   10
5 3
dx
6 
6
2

d
dx
x

h  sin x   5 3
6
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 178
Use the table of values to calculate the derivative of the function
at the given point.
x
1 4 6
81. g x , x  16
f  x 4 0 6
 
d
g
dx
 
 
 1 
x  g x 

2
x


 1 
 g  16 

2
16


1
1 1 1

 g  4 

8
8 2 16
 
d
g
dx
 x
x 16
f  x 5
g  x
7
4
4 1
6
1
2
3
g  x 5
1

16
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 178
Compute the indicated higher derivatives.
3
d
11
85.
 3x  9 
dx3
d
11
10
10
 3x  9   11 3x  9   3  33  3x  9 
dx
d2
d
11
10
9
9
3
x

9

33
3
x

9

33
10
3
x

9
3

990
3
x

9



 
 


2 
dx
dx
d3
d
11
9
8
8
3 x  9   990  3 x  9   990  9  3 x  9   3   26730  3 x  9 
3 
dx
dx
d3
11
8
3 x  9   26730  3 x  9 
3 
dx
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 178
89. Compute the second derivative of sin  g  x   at x  2, assuming

that g  2  
4
, g   2   5, and g   2   3.
d
d
sin  g  x    sin u  cos u u 
dx
dx
d2
d
  u    sin u  u 


sin
u

cos
u
u

cos
u
u


dx 2
dx
 2
2
22 2
2
2
 u  cos u   u   sin u  3 

5


   

2
2
2




d2
sin u  11 2
2
dx
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 178
93. The force F (Newtons) between two charged particles is F =
100/r2, where r is the distance (meters) between them. Find
dF/dt at t = 10 if distance at time t (sec) is r = 1 + 0.4t2.
F  100r 2  r  1  0.4t 2  r   0.8t
dF
200
200
3


 100  2  r r  3 r 
0.8t 
3 
2
dt
r
1  0.4t 
dF
dt
dF
dt

t 10
200
1  0.4 10 
2 3
 0.8 10   
1600
 0.9518
3
41
 0.9518 N/s
t 10
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Jon Rogawski
Calculus, ET
First Edition
Chapter 3: Differentiation
Section 3.8: Implicit Differentiation
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
The graph in Figure 1 is not that of a function, but of a relation. It
is said that y is defined implicitly, as it could be difficult or
impossible to solve the equation for y. To find the slope of the
tangent line, we must use implicit differentiation.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Consider the unit
circle shown in Figure 2.
How do we determine
the slope at the point
in question? Consider the
equation for the unit
circle x2 + y2 = 1.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
The relation y4 + xy = x3 – x + 2 may be broken down into two
functions, each defining a branch of the curve and neither
violating the vertical line rule. The point where the branches split
is the point at which the curve has a vertical slope.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Find the equation of the tangent to the curve y4 + xy = x3 – x + 2
at (1, 1).
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
dy
t2
Find
where cos  ty  
dt
y
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Find the slope of the tangent line to the
curve e xy  x  y at the point 1,0  .
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework



Homework Assignment #17
Read Section 3.9
Page 184, Exercises: 1 – 49 (EOO)
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company