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DIFFERENTIATION RULESDIFFERENTIATION RULES
3
Before starting this section,
you might need to review the
trigonometric functions.
DIFFERENTIATION RULES
In particular, it is important to remember that,
when we talk about the function f defined for
all real numbers x by f(x) = sin x, it is
understood that sin x means the sine of
the angle whose radian measure is x.
DIFFERENTIATION RULES
A similar convention holds for
the other trigonometric functions
cos, tan, csc, sec, and cot.
Recall from Section 2.5 that all the trigonometric functions are continuous at every number in their domains.
DIFFERENTIATION RULES
DIFFERENTIATION RULES
3.6Derivatives of
Trigonometric Functions
In this section, we will learn about:
Derivatives of trigonometric functions
and their applications.
Let’s sketch the graph of the function
f(x) = sin x and use the interpretation of f’(x)
as the slope of the tangent to the sine curve
in order to sketch the graph of f’.
DERIVATIVES OF TRIGONOMETRIC FUNCTIONS
Then, it looks as if the graph of f’ may
be the same as the cosine curve.
DERIVATIVES OF TRIGONOMETRIC FUNCTIONS
Let’s try to confirm
our guess that, if f(x) = sin x,
then f’(x) = cos x.
DERIVATIVES OF TRIGONOMETRIC FUNCTIONS
From the definition of a derivative, we have:
0 0
0
0
0
0 0 0
( ) ( ) sin( ) sin'( ) lim lim
sin cos cos sin h sinlim
sin cos sin cos sinlim
cos 1 sinlim sin cos
cos 1limsin lim lim cos lim
h h
h
h
h
h h h h
f x h f x x h xf x
h hx h x x
hx h x x h
h h
h hx x
h h
hx x
h
0
sin h
h
DERIVS. OF TRIG. FUNCTIONS Equation 1
Two of these four limits are easy to
evaluate.
DERIVS. OF TRIG. FUNCTIONS
0 0 0 0
cos 1 sinlimsin lim lim cos limh h h h
h hx x
h h
Since we regard x as a constant
when computing a limit as h → 0,
we have:
DERIVS. OF TRIG. FUNCTIONS
limh 0
sin x sin x
limh 0
cos x cos x
The limit of (sin h)/h is not so obvious.
In Example 3 in Section 2.2, we made
the guess—on the basis of numerical and
graphical evidence—that:
0
sinlim 1
DERIVS. OF TRIG. FUNCTIONS Equation 2
We now use a geometric argument
to prove Equation 2.
Assume first that θ lies between 0 and π/2.
DERIVS. OF TRIG. FUNCTIONS
The figure shows a sector of a circle with
center O, central angle θ, and radius 1.
BC is drawn perpendicular to OA.
By the definition of radian measure, we have arc AB = θ.
Also, |BC| = |OB| sin θ = sin θ.
DERIVS. OF TRIG. FUNCTIONS Proof
sinsin so 1
DERIVS. OF TRIG. FUNCTIONS
We see that
|BC| < |AB| < arc AB
Thus,
Proof
Let the tangent lines at A and B
intersect at E.
DERIVS. OF TRIG. FUNCTIONS Proof
You can see from this figure that
the circumference of a circle is smaller than
the length of a circumscribed polygon.
So,
arc AB < |AE| + |EB|
DERIVS. OF TRIG. FUNCTIONS Proof
Thus,
θ = arc AB < |AE| + |EB|
< |AE| + |ED|
= |AD| = |OA| tan θ
= tan θ
DERIVS. OF TRIG. FUNCTIONS Proof
Therefore, we have:
So,
sin
cos
DERIVS. OF TRIG. FUNCTIONS
sincos 1
Proof
We know that .
So, by the Squeeze Theorem,
we have:
0 0lim1 1 and lim cos 1
0
sinlim 1
DERIVS. OF TRIG. FUNCTIONS Proof
However, the function (sin θ)/θ is an even
function.
So, its right and left limits must be equal.
Hence, we have:0
sinlim 1
DERIVS. OF TRIG. FUNCTIONS Proof
We can deduce the value of the remaining
limit in Equation 1 as follows.
0
0
2
0
cos 1lim
cos 1 cos 1lim
cos 1
cos 1lim
(cos 1)
DERIVS. OF TRIG. FUNCTIONS
2
0
0
0 0
0
sinlim
(cos 1)
sin sinlim
cos 1
sin sin 0lim lim 1 0
cos 1 1 1
cos 1lim 0
DERIVS. OF TRIG. FUNCTIONS Equation 3
If we put the limits (2) and (3) in (1),
we get:
0 0 0 0
cos 1 sin'( ) limsin lim lim cos lim
(sin ) 0 (cos ) 1
cos
h h h h
h hf x x x
h hx x
x
DERIVS. OF TRIG. FUNCTIONS
So, we have proved the formula for
the derivative of the sine function:
(sin ) cosd
x xdx
DERIV. OF SINE FUNCTION Formula 4
Differentiate y = x2 sin x.
Using the Product Rule and Formula 4, we have:
2 2
2
(sin ) sin ( )
cos 2 sin
dy d dx x x x
dx dx dx
x x x x
Example 1DERIVS. OF TRIG. FUNCTIONS
Using the same methods as in
the proof of Formula 4, we can prove:
(cos ) sind
x xdx
Formula 5DERIV. OF COSINE FUNCTION
The tangent function can also be
differentiated by using the definition
of a derivative.
However, it is easier to use the Quotient Rule
together with Formulas 4 and 5—as follows.
DERIV. OF TANGENT FUNCTION
2
2
2 22
2 2
2
sin(tan )
cos
cos (sin ) sin (cos )
coscos cos sin ( sin )
cos
cos sin 1sec
cos cos
(tan ) sec
d d xx
dx dx x
d dx x x x
dx dxx
x x x x
x
x xx
x xd
x xdx
DERIV. OF TANGENT FUNCTION Formula 6
The derivatives of the remaining
trigonometric functions—csc, sec, and cot—
can also be found easily using the Quotient
Rule.
DERIVS. OF TRIG. FUNCTIONS
We have collected all the differentiation
formulas for trigonometric functions here. Remember, they are valid only when x is measured
in radians.
2 2
(sin ) cos (csc ) csc cot
(cos ) sin (sec ) sec tan
(tan ) sec (cot ) csc
d dx x x x x
dx dxd d
x x x x xdx dxd d
x x x xdx dx
DERIVS. OF TRIG. FUNCTIONS
Differentiate
For what values of x does the graph of f
have a horizontal tangent?
sec( )
1 tan
xf x
x
Example 2DERIVS. OF TRIG. FUNCTIONS
The Quotient Rule gives:
2
2
2
2 2
2
2
(1 tan ) (sec ) sec (1 tan )'( )
(1 tan )
(1 tan )sec tan sec sec
(1 tan )
sec (tan tan sec )
(1 tan )
sec (tan 1)
(1 tan )
d dx x x x
dx dxf xx
x x x x x
x
x x x x
x
x x
x
Example 2DERIVS. OF TRIG. FUNCTIONS
In simplifying the answer,
we have used the identity
tan2 x + 1 = sec2 x.
DERIVS. OF TRIG. FUNCTIONS Example 2
Since sec x is never 0, we see that f’(x)
when tan x = 1. This occurs when x = nπ + π/4,
where n is an integer.
Example 2DERIVS. OF TRIG. FUNCTIONS
Trigonometric functions are often used
in modeling real-world phenomena.
In particular, vibrations, waves, elastic motions, and other quantities that vary in a periodic manner can be described using trigonometric functions.
In the following example, we discuss an instance of simple harmonic motion.
APPLICATIONS
An object at the end of a vertical spring
is stretched 4 cm beyond its rest position
and released at time t = 0. In the figure, note that the downward
direction is positive. Its position at time t is
s = f(t) = 4 cos t Find the velocity and acceleration
at time t and use them to analyze the motion of the object.
Example 3APPLICATIONS
The velocity and acceleration are:
(4cos ) 4 (cos ) 4sin
( 4sin ) 4 (sin ) 4cos
ds d dv t t t
dt dt dt
dv d da t t t
dt dt dt
Example 3APPLICATIONS
The object oscillates from the lowest point
(s = 4 cm) to the highest point (s = -4 cm).
The period of the oscillation
is 2π, the period of cos t.
Example 3APPLICATIONS
The speed is |v| = 4|sin t|, which is greatest
when |sin t| = 1, that is, when cos t = 0.
So, the object moves fastest as it passes through its equilibrium position (s = 0).
Its speed is 0 when sin t = 0, that is, at the high and low points.
Example 3APPLICATIONS
The acceleration a = -4 cos t = 0 when s = 0.
It has greatest magnitude at the high and
low points.
Example 3APPLICATIONS
Find the 27th derivative of cos x.
The first few derivatives of f(x) = cos x are as follows:
(4)
(5)
'( ) sin
''( ) cos
'''( ) sin
( ) cos
( ) sin
f x x
f x x
f x x
f x x
f x x
Example 4DERIVS. OF TRIG. FUNCTIONS
We see that the successive derivatives occur in a cycle of length 4 and, in particular, f (n)(x) = cos x whenever n is a multiple of 4.
Therefore, f (24)(x) = cos x
Differentiating three more times, we have:
f (27)(x) = sin x
Example 4DERIVS. OF TRIG. FUNCTIONS
Our main use for the limit in Equation 2
has been to prove the differentiation formula
for the sine function.
However, this limit is also useful in finding certain other trigonometric limits—as the following two examples show.
DERIVS. OF TRIG. FUNCTIONS
Find
In order to apply Equation 2, we first rewrite the function by multiplying and dividing by 7:
0
sin 7lim
4x
x
x
sin 7 7 sin 7
4 4 7
x x
x x
Example 5DERIVS. OF TRIG. FUNCTIONS
If we let θ = 7x, then θ → 0 as x → 0.
So, by Equation 2, we have:
0 0
0
sin 7 7 sin 7lim lim
4 4 7
7 sinlim
4
7 71
4 4
x x
x x
x x
Example 5DERIVS. OF TRIG. FUNCTIONS
Calculate .
We divide the numerator and denominator by x:
by the continuity of cosine and Eqn. 2
0lim cotx
x x
Example 6DERIVS. OF TRIG. FUNCTIONS
0 0 0
0
0
cos coslim cot lim lim
sinsin
lim cos cos0sin 1lim
1
x x x
x
x
x x xx x
xxx
x
x
x