5128_Ch03_pp098-184

152 Chapter 3 Derivatives
EXAMPLE 7 Finding Slope
(a) Find the slope of the line tangent to the curve y sin 5 x at the point where
x p3.
(b) Show that the slope of every line tangent to the curve y 11 2x 3 is positive.
SOLUTION
(a) d y
5 sin
dx
4 d
x • sin x Power Chain Rule with u sin x, n 5
d x
5 sin 4 x cos x
The tangent line has slope
d 4
y
dx
|xp3
5( 3
2
) ( 1
2 ) 4 5
.
32
(b)
d y d
1 2x
dx
d x
3
31 2x 4 d
• 1 2x d x
31 2x 4 • 2
1
62x 4
Power Chain Rule with
u (1 2x), n 3
At any point x, y on the curve, x 12 and the slope of the tangent line is
d y
,
dx
1
62x 4
the quotient of two positive numbers. Now try Exercise 53.
EXAMPLE 8
Radians Versus Degrees
It is important to remember that the formulas for the derivatives of both sin x and cos x
were obtained under the assumption that x is measured in radians, not degrees. The
Chain Rule gives us new insight into the difference between the two. Since 180° p
radians, x° px180 radians. By the Chain Rule,
d
d
sin x° sin d x d x ( px
18
) 0
p
180 ( cos px
18
) 0
p
cos x°.
180 See Figure 3.44.
The factor p180, annoying in the first derivative, would compound with repeated differentiation.
We see at a glance the compelling reason for the use of radian measure.
1
y
y sin (x°) sin x ___
180
x
y sin x
180
Figure 3.44 sin x°) oscillates only p180 times as often as sin x oscillates. Its maximum slope
is p180. (Example 8)