Section 2.4 I)erivatives of Trigonometric Functions
SOLUTION
The derivative of y
=
tan x is
derivative is equal to sec2
=
(2)’
dy
=
2 x. When x
sec
=
117
/4, the
2. us the required line has slope 2 and
passes through (/4, 1). Thus
y
1
—
2(x
—
y
• EXAMPLE 8
line is horizontal.
I
2x
—
+ 1
Find all points on the graph of y
=
2 x where the tangent
sin
SOLUTION The tangent line is horizontal when the derivative is equal to zero.
To get the derivative of sin
2 x, we use the Product Rule.
(1.,
—-sin x
cix
=
ci
—(sm x sin x)
dx
sin x cos x + sin x cos x
=
2 sin x cos x
The product of sin x and cos x is equal to zero when either sin x or cos x is equal to
3
zero; that is, at x
0, ±—, +z, ±
(oncepts Review
I. By definition, D(sin x)
=
The two displayed limits have the values
lim
2. To evaluate the limit in the preceding statement, we Iirst
he Addition Identity for the sine function and then do a little
Hhra to obtain
I) (sin x)
(—sin
)( Ii
1iO
1
—
C05
11
h
+
sin h
(cos x)(lim)
h—()
ii
\
and
respectively.
3. The result of the calculation in the preceding statement is
the important derivative formula D (sin x) =
The corre
sponding derivative formula D(cos x) =
is obtained in
similar manner.
.
4. At x = /3, D(sin x) has the value
Thus, the
equation of the tangent line to v = sin x at x = /3 is
.
Problem Set 2.4
Iii I roblerns 1—18, find Dry.
I. y
3. y
=
=
2 sin x + 3 cos x
x + cos
2
sin
x
2
5. y=secx= t/cosx
7. y
9. y
=
—
tan x
=
sin x
cos x
sin x + cos x
cos x
-______________
II. )‘=sinxcosx
sinx
13. y=—
x
15. y
=
Cosx
2
x
17. y
2. y
4. y
=
=
2x
sin
x
1 — cos
x
2
6. y=cscx=1/sinx
cos x
8. ycotx=—
sin x
10. y
sin x + cos x
=
16. y
1— cosx
x
x cos x + sin x
=
-—----x- + 1
2x
tan
18. y
=
sec’ x
19. Find the equation of the tangent line to y
1.
-—
=
cos x at
=
cot x at
=
20. Find the equation of the tangent line to y
x=i.
21. Use the trigonometric identity sin 2x
along with the Product Rule to find D sin 2x.
22. Use the trigonometric identity cos 2x
along with the Product Rule to find D cos 2x.
tail x
12. y=sinxtanx
14. y=
=
=
2 sin x cos x
=
2x — 1
2 cos
23. A Ferris wheel of radius 30 feet is rotating counterclock
wise with an angular velocity of 2 radians per second. How fast is
a seat on the rim rising (in the vertical direction) when it is 15 feet
above the horizontal line through the center of the wheel? hint:
Use the result of Problem 21.