d
dt
(5

sin 2
t
)
g
/
s
t
d
=
d
dt
(tan (5

sin 2
t
))
5

sin 2
t
,
g
s
t
d
=
tan
s
5

sin 2
t
d
.
Derivative of tan
u
with
u
=
5

sin 2
t
Derivative of
with
u
=
2
t
5

sin
u
H
ISTORICAL
B
IOGRAPHY
Johann Bernoulli
(1667–1748)
Power Chain Rule with
u
=
5
x
3

x
4
,
n
=
7
Power Chain Rule with
u
=
3
x

2,
n
= 
1
because sin
n
x
means
s
sin
x
d
n
,
n
, 
1.
Chapter 2:
Differentiation
109
02 C2 Differentiation.indd
109
7/27/12
1:19:24 PM
We have textbook solutions for you!
The document you are viewing contains questions related to this textbook.
The document you are viewing contains questions related to this textbook.
Expert Verified
EXAMPLE 6
In Section
2
.2, we saw that the absolute value function
is not
differentiable at
x
#
0. However, the function
is
differentiable at all other real numbers as
we now show. Since
, we can derive the following formula:
EXAMPLE 7
Show that the slope of every line tangent to the curve
is
positive.
Solution
We find the derivative:
Power Chain Rule with
At any point (
x
,
y
) on the curve,
and the slope of the tangent line is
the quotient of two positive numbers.
EXAMPLE 8
The formulas for the derivatives of both sin
x
and cos
x
were obtained un
der the assumption that
x
is measured in radians,
not
degrees. The Chain Rule gives us new
insight into the difference between the two. Since
radians,
radi
ans where
x
° is the size of the angle measured in degrees.
By the Chain Rule,
See Figure
2
.25. Similarly, the derivative of
The factor
would compound with repeated differentiation. We see here the
advantage for the use of radian measure in computations.
p
>
180
cos
s
x
°
d
is

s
p
>
180
d
sin
s
x
°
d
.
d
dx
sin
s
x
°
d
=
d
dx
sin
(
p
x
180
)
=
p
180
cos
(
p
x
180
)
=
p
180
cos
s
x
°
d
.
x
°
=
p
x
>
180
180°
=
p
dy
dx
=
6
s
1

2
x
d
4
,
x
,
1
>
2
=
6
s
1

2
x
d
4
= 
3
s
1

2
x
d

4
!
s

2
d
u
=
s
1

2
x
d
,
n
= 
3
= 
3
s
1

2
x
d

4
!
d
dx
s
1

2
x
d
dy
dx
=
d
dx
s
1

2
x
d

3
y
=
1
>
s
1

2
x
d
3
=
x
0
x
0
,
x
,
0.
=
1
2
0
x
0
!
2
x
=
1
2
2
x
2
!
d
dx
(
x
2
)
d
dx
(
0
x
0
)
=
d
dx
2
x
2
0
x
0
=
2
x
2
y
=
0
x
0
Power Chain Rule with
u
=
x
2
,
n
=
1
>
2,
x
,
0
2
x
2
=
0
x
0
Derivative of the
Absolute Value Function
d
dx
(
0
x
0
)
=
x
0
x
0
,
x
,
0
x
y
1
180
y
&
sin
x
y
&
sin(
x
$
)
&
sin
�
x
180
FIGURE
2
.25
oscillates only
times as often as
oscillates. Its
maximum slope is
at
(Example 8).
x
=
0
p
>
180
sin
x
p
>
180
Sin
s
x
°
d
110
Part 1:
Calculus
02 C2 Differentiation.indd
110
7/27/12
1:19:33 PM
made of metal, its length will vary with temperature, either in
creasing or decreasing at a rate that is roughly proportional to
L
.
In symbols, with
u
being temperature and
k
the proportionality
constant,
Assuming this to be the case, show that the rate at which the pe
riod changes with respect to temperature is
.