i=14
Differentiate.
1.
flx)
=
x%sinx
3.
f(x)
=e*cosx
N
.
f(x)
=xcosx
+
2tanx
FY
.
y=128ecx
—
¢scx
5.
y
=
secf
tanf
6.
g(6)
=
e'(tand
—
@)
cot
¢
7.
y=ccost+
t%sin¢t
8.
f(t)=7
X
S
y=—
10.
y
=
sin8
cos8
+
2
—tanx
P
)
sin
6
*
cos
X
11,
f(6)
=
———
12,
y=
————
1®
1
+
cos
6
¥
1
—sinx
tsint
sin
¢
13.
y
=
4
y=——
AT
YT
T
¥
tans
15.
f(6)
=
0
cos
6
sin
§
16.
f(t)
=
te'cot
t
d
17.
Prove
thatd—(csc
X)
=
—csc
x
cot
x.
x
d
18.
Prove
that
"
(sec
x)
=
sec
x
tan
x.
d
19.
Prove
that
=
(cot
x)
=
—csc'x.
20.
Prove,
using
the
definition
of
derivative,
that
if
S(x)
=
cos
x,
then
f'(x)
=
—sin
x.
272%
Find
an
equation
of
the
tangent
line
to
the
curve
at
the
given
point,
21.
y
=sinx
+
cosx,
(0,
1)
Zzoy=e'cosx,
(0,1)
(7T,
_1)
23.
y
=
cos
x
—
sin
x,
24,
y=x+tanx,
(mm)
25.
(a)
Find
an
equation
of
the
tangent
line
to
the
curve
¥
=
2x
sin
x
at
the
point
(7/2,
).
(b)
Illustrate
part
(a)
by
graphing
the
curve
and
the
tangent
line
on
the
same
screen.
26.
(a)
Find
an
equation
of
the
tangent
line
to
the
curve
¥
=
3x
+
6
cos
x
at
the
point
(7/3,
77
+
3).
(b)
Ulustrate
part
(a)
by
graphing
the
curve
and
the
tangent
line
on
the
same
screen.
27.
(a)
If
f(x)
=
secx
—
x,
find
f'(x).
;
(b)
Check
to
see
that
your
answer
to
part
(a)
is
reasonable
by
graphing
both
f
and
("
for

x
<
/2.
28.
(a)
If
f(x)
=
e"cosx,
find
f'(x)
and
f"(x).
(b)
Check
to
see
that
your
answers
to
part
(a)
are
reasonable
by
graphing
f,
/',
and
f".
20
IEH(8)
=
6
sind,
find
H'(6)
and
H"
().
30.
If
f(1)
=
sect,
find
f"(/4).
31.
(a)
Use
the
Quotient
Rule
to
differentiate
the
function
_
tanx
—
]
sec
x
(b)
Simplify
the
expression
for
f(x)
by
writing
it
in
terms
of
sin
x
and
cos
x,
and
then
find
f'(x).
(¢)
Show
that
your
answers
to
parts
(a)
and
(b)
are
equivalent,
32.
Suppose
f(m/3)
=
4
and
f'(7/3)
=
—2, and
let
g(x)
=
f(x)
sin
x
and
/1(x)
=
(cos
x)/f(x).
Find
(@)
¢'(7/3)
(b)
#'(m/3)
3334
Por
what
values
of
x
does
the
graph
of
f
have
a
horizon
tal
tangent?
35
f(x)
=x
+
2sinx
34,
f(x)
=e'cosx
35.
A
mass
on
a
spring
vibrates
horizontally
on
a
smooth
level
surface
(see
the
figure).
lis
equation
of
motion
is
x(z)
=
8sin
1,
where
t
is
in
seconds
and
x
in
centimeters.
(a)
Find
the
velocity
and
acceleration
at
time
.
(b)
Find
the
position,
velocity,
and
acceleration
of
the
mass
at
time
£
=
24r/3.
In
what
direction
is
it
moving
at
that
time?
]
equilibrium

posilion

s
0
X
/"
36.
An
elastic
band
is
hung
on
a
hook
and
a
mass
is
hung
on
the
lower
end
of
the
band.
When
the
mass
is
pulled
downward
and
then
released,
it
vibrates
vertically.
The
equation
of
motion
is
s
=
2
cos
¢t
+
3sint,
¢
=
0,
where
s
is
measured
in
centimeters
and
7
in
seconds.
(Take
the
positive
direction
to
be
downward.)
(a)
Find
the
velocity
and
acceleration
at
time
¢.
(b)
Graph
the
velocity
and
acceleration
functions.
(c)
When
does
the
mass
pass
through
the
equilibrium
position
for
the
first
time?
(d)
How
far
from
its
equilibrium
position
does
the
mass
travel?
(e)
When
is
the
speed
the
greatest?
/.
Aladder
10
ft
long
rests
against
a
vertical
wall.
Let
8
be
the
angle
between
the
top
of
the
ladder
and
the
wall
and
let
x
be
the
distance
from
the
bottom
of
the
ladder
to
the
wall.
If
the
bottom
of
the
ladder
slides
away
from
the
wall,
how
fast
does
x
change
with
respect
to
§
when
8
=
7
/39
38.
An
object
with
weight
W
is
dragged
along
a
horizontal
plane
by
a
force
acting
along
a
rope
attached
to
the
object.