Download UNIT 3 Lesson 3 Derivatives of Sine and Cosine

UNIT 3 Lesson 3
Derivatives of Sine and Cosine
1
DERIVATIVE OF THE SINE FUNCTION
𝒚 = 𝐬𝐢𝐧 𝒙
𝒅𝒚
=?
𝒅𝒙
2
Find the derivative of f(x) = sin x from first principles:
𝐬𝐢𝐧 𝒙 + 𝒉 − 𝐬𝐢𝐧 𝒙
𝒉→𝟎
𝒙+𝒉 −𝒙
𝒇′ 𝒙 = 𝐥𝐢𝐦
Sum Identity: sin (A + B) = sin A cos B + cos A sin B
𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒉 + 𝐜𝐨𝐬 𝒙 𝐬𝐢𝐧 𝒉 − 𝐬𝐢𝐧 𝒙
𝒉→𝟎
𝒉
𝒇′ 𝒙 = 𝐥𝐢𝐦
𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒉 − 𝐬𝐢𝐧 𝒙 + 𝐜𝐨𝐬 𝒙 𝐬𝐢𝐧 𝒉
𝒉→𝟎
𝒉
𝒇′ 𝒙 = 𝐥𝐢𝐦
𝐬𝐢𝐧 𝒙 (𝐜𝐨𝐬 𝒉 − 𝟏)) + 𝐜𝐨𝐬 𝒙 𝐬𝐢𝐧 𝒉
𝒉→𝟎
𝒉
𝒇′ 𝒙 = 𝐥𝐢𝐦
𝒇′
𝐬𝐢𝐧 𝒙 (𝐜𝐨𝐬 𝒉 − 𝟏))
𝐜𝐨𝐬 𝒙 𝐬𝐢𝐧 𝒉
𝒙 = 𝐥𝐢𝐦
+ 𝐥𝐢𝐦
𝒉→𝟎
𝒉
𝒉→𝟎
𝒉
𝐜𝐨𝐬 𝒉 − 𝟏
𝐬𝐢𝐧 𝒉
+ 𝐥𝐢𝐦 𝐜𝐨𝐬 𝒙 𝐥𝐢𝐦
𝒉→𝟎
𝒉→𝟎
𝐡→𝟎 𝒉
𝒉
𝒇′ 𝒙 = 𝐥𝐢𝐦 𝐬𝐢𝐧 𝒙 × 𝐥𝐢𝐦
𝒉→𝟎
continued
3
𝐜𝐨𝐬 𝒉
𝐬𝐢𝐧 𝒉
𝒇 𝒙 = 𝐬𝐢𝐧 𝒙 𝐥𝐢𝐦
+ 𝐜𝐨𝐬 𝒙 𝐥𝐢𝐦
𝒉→𝟎 𝒉
𝒉→𝟎 𝒉
′
From the limits that we developed in the last lesson, we know
that
𝐜𝐨𝐬 𝒉 − 𝟏
𝐬𝐢𝐧 𝒉
𝐥𝐢𝐦
=𝟎
𝐥𝐢𝐦
=𝟏
𝒉→𝟎
𝒉→𝟎 𝒉
𝒉
𝒇′ 𝒙 = 𝐬𝐢𝐧 𝒙 𝟎 + 𝐜𝐨𝐬 𝒙(𝟏)
𝒇′ 𝒙 = 𝐜𝐨𝐬 𝒙
Our first trigonometric derivative is:
𝒅
𝐬𝐢𝐧 𝒙 = 𝐜𝐨𝐬 𝒙
𝒅𝒙
4
EXAMPLE 1: Find the derivative of 𝒇 𝒙 = 𝟑 𝐬𝐢𝐧 𝒙
𝒇′
𝒙 = 𝟑 𝐜𝐨𝐬 𝒙
𝒔𝒊𝒏 𝜽 =
𝒚
=𝒚
𝟏
𝒄𝒐𝒔 𝜽 =
𝒙
=𝒙
𝟏
Find the slope of the tangent at
𝒕𝒂𝒏 𝜽 =
,
𝝅
𝒙=
𝟑
𝝅
𝒙=
𝟐
𝝅
𝝅
𝟏
𝟑
𝒇
= 𝟑 𝐜𝐨𝐬
=𝟑
=
𝟑
𝟑
𝟐
𝟐
𝝅
𝝅
′
𝒇
= 𝟑 𝐜𝐨𝐬
= 𝟑(𝟎) = 𝟎
𝟐
𝟐
𝒙=𝝅
𝒇′ 𝝅 = 𝟑 𝐜𝐨𝐬 𝝅 = 𝟑 −𝟏 = −𝟑
𝒚
𝒙
′
5
EXAMPLE 2: Find the derivative of 𝒈 𝒙 = 𝐬𝐢𝐧(𝟑𝒙)
Use the Chain Rule
𝒈′ 𝒙 = (𝐜𝐨𝐬(𝟑𝒙))(𝟑) = 𝟑 𝐜𝐨𝐬(𝟑𝒙)
Find the slope of the tangent at
𝒙=𝟎
𝒈′ 𝟎 = 𝟑 𝐜𝐨𝐬 𝟑 × 𝟎 = 𝟑 cos 𝟎 = 𝟑 𝟏 = 𝟑
𝝅
𝒙=
𝟔
𝒈′
𝝅
𝝅
𝝅
= 𝟑 𝐜𝐨𝐬 𝟑 ×
= 𝟑 𝐜𝐨𝐬 = 𝟑 𝟎 = 𝟎
𝟔
𝟔
𝟐
𝝅
𝒙=
𝟑
𝒈′
𝝅
𝝅
= 𝟑 𝐜𝐨𝐬 𝟑 ×
= 𝟑 𝐜𝐨𝐬 𝝅 = 𝟑 −𝟏 = −𝟑
𝟑
𝟑
6
EXAMPLE 3: Differentiate h(x) = – 2sin(5x)
𝒉′(𝒙) = −𝟐 𝐜𝐨𝐬(𝟓𝒙)(𝟓)
𝒉′(𝒙) = −𝟏𝟎 𝐜𝐨𝐬(𝟓𝒙)
.
Find the slope of the tangent at
𝝅
𝒙=
𝟓
𝝅
𝝅
𝒉′
= −𝟏𝟎 𝐜𝐨𝐬 𝟓 ×
= −𝟏𝟎 𝐜𝐨𝐬 𝝅 = −𝟏𝟎 −𝟏 = 𝟏𝟎
𝟓
𝟓
7
EXAMPLE 4: Find the derivative of 𝒚 = 𝐬𝐢𝐧 𝒙
𝟐
Find the slope of the tangent at
𝒙=𝝅
𝒅𝒚
= 𝟐 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒙
𝒅𝒙
𝒅𝒚
= 𝟐 𝐬𝐢𝐧 𝝅 𝐜𝐨𝐬 𝝅
𝒅𝒙
𝒅𝒚
= 𝟐 −𝟏 𝟎 = 𝟎
𝒅𝒙
8
EXAMPLE 5: Find the derivative of 𝒚 = 𝟑 𝐬𝐢𝐧𝟐 𝒙 = 𝟑 𝐬𝐢𝐧 𝒙
𝒅𝒚
= 𝟔 𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒙
𝒅𝒙
Find the slope of the tangent at
𝒙=𝝅
𝟔 𝐬𝐢𝐧 𝝅 𝐜𝐨𝐬 𝝅 = 𝟔 𝟎 −𝟏 = 𝟎
9
𝟐
EXAMPLE 6: Find the derivative of 𝑦 = sin2 3𝑥 = sin(3𝑥)
2
𝒅𝒚
= 𝟐 𝐬𝐢𝐧(𝟑𝒙)(𝟑) 𝐜𝐨𝐬(𝟑𝒙)
𝒅𝒙
𝒅𝒚
= 𝟔 𝐬𝐢𝐧(𝟑𝒙) 𝐜𝐨𝐬(𝟑𝒙)
𝒅𝒙
Find the slope of the tangent at
𝝅
𝒅𝒚
𝝅
𝝅
𝒙=
= 𝟔 𝐬𝐢𝐧(𝟑 × ) 𝐜𝐨𝐬(𝟑 × )
𝟏𝟐
𝒅𝒙
𝟏𝟐
𝟏𝟐
𝒅𝒚
𝝅
𝝅
= 𝟔 𝐬𝐢𝐧 𝐜𝐨𝐬
𝒅𝒙
𝟒
𝟒
𝟏
𝟏
𝟔
𝟔×
×
= =𝟑
𝟐
𝟐 𝟐
10
EXAMPLE 7: Differentiate 𝑓 𝑥 = 4 sin2 𝑥 2 + 2
𝒇 𝒙 = 𝟒 𝒔𝒊𝒏𝟐 𝒙𝟐 + 𝟐
𝟐
𝒇 𝒙 = 𝟒 𝐬𝐢𝐧 𝒙 + 𝟐
REWRITE
𝟐
Using the Chain Rule
→
𝒇′ 𝒙 = 𝟐(𝟒) 𝐬𝐢𝐧 𝒙𝟐 + 𝟐
𝐜𝐨𝐬 𝒙𝟐 + 𝟐 (𝟐𝒙)
𝒇′ 𝒙 = 𝟏𝟔𝒙 𝐬𝐢𝐧 𝒙𝟐 + 𝟐
𝐜𝐨𝐬 𝒙𝟐 + 𝟐
Using the sine double angle identity
2sin A cos A = sin 2A
𝒇′ 𝒙 = 𝟖𝒙 𝟐 𝐬𝐢𝐧 𝒙𝟐 + 𝟐
𝐜𝐨𝐬 𝒙𝟐 + 𝟐
𝒇′ 𝒙 = 𝟖𝒙 𝐬𝐢𝐧 𝟐 𝒙𝟐 + 𝟐
11
DERIVATIVE OF THE COSINE FUNCTION
What is
d
cos x
dx
?
12
To find the derivative of the function f(x) = cos x, we can
use a complementary angle identity in combination
with the derivative of f(x) = sin x.
COMPLEMENTARY ANGLES (sum is
A
90o
or
𝝅
)
𝟐
c
b
B
C
a
Angles A and B are complementary angles.
𝝅
𝝅
A= –B
and B = – A
𝟐
𝟐
𝒂
𝒂
𝝅
𝐜𝐨𝐬 𝑩 = = 𝐬𝐢𝐧 𝑨 = = 𝐬𝐢𝐧( – 𝑩)
𝒄
𝒄
𝟐
𝒂
𝒂
𝝅
𝐬𝐢𝐧 𝑨 = = 𝒄𝒐𝒔 𝑩 = = 𝒄𝒐𝒔( – 𝑨)
𝒄
𝒄
𝟐
13
Find the derivative of 𝒚 = 𝐜𝐨𝐬 𝒙
𝝅
𝒚 = 𝐜𝐨𝐬 𝒙 = 𝐬𝐢𝐧
−𝒙
𝟐
We now differentiate using the Chain Rule:
𝑑𝑦
𝜋
= cos − 𝑥 (−1)
𝑑𝑥
2
𝑑𝑦
𝜋
= −cos − 𝑥
𝑑𝑥
2
𝑑𝑦
= − sin 𝑥
𝑑𝑥
This is the second trigonometric derivative:
𝒅
𝐜𝐨𝐬 𝒙 = − 𝐬𝐢𝐧 𝒙
𝒅𝒙
14
EXAMPLE 8: Differentiate f(x) = sin (x) cos (x) and use the
derivative to find the slope of the tangent to the function when
𝜋
𝑥=
3
Using the Product Rule
𝒇′ 𝒙 = 𝐬𝐢𝐧 𝒙 − 𝐬𝐢𝐧 𝒙 + 𝐜𝐨𝐬 𝒙 𝐜𝐨𝐬 𝒙
2
3
𝒇′ 𝒙 = − 𝐬𝐢𝐧𝟐 𝒙 + 𝐜𝐨𝐬𝟐 𝒙
𝒇′
𝒙 =
− 𝐬𝐢𝐧𝟐
𝝅
𝒇′
=−
𝟑
𝟑
𝟐
𝝅
𝝅
𝟐
+ 𝐜𝐨𝐬
𝟑
𝟑
𝟐
𝟏
+
𝟐
𝟐
1
𝟑 𝟏
𝟏
=− + =−
𝟒 𝟒
𝟐
15
EXAMPLE 9: Differentiate 𝑓 𝑥 =
sin 3𝑥
cos 𝑥 2
Using the Quotient Rule and Chain Rule
𝟐 𝟑 𝐜𝐨𝐬 𝟑𝒙 − 𝐬𝐢𝐧 𝟑𝒙 − 𝐬𝐢𝐧 𝒙𝟐 (𝟐𝒙)
𝐜𝐨𝐬
𝒙
𝒇′ 𝒙 =
𝐜𝐨𝐬 𝒙𝟐 𝟐
𝟐 )(𝐜𝐨𝐬 𝟑𝒙) + 𝟐𝒙(𝐬𝐢𝐧 𝟑𝒙) 𝐬𝐢𝐧 𝒙𝟐
𝟑(𝐜𝐨𝐬
𝒙
𝒇′ 𝒙 =
𝐜𝐨𝐬 𝒙𝟐 𝟐
16
2
2
EXAMPLE 10: Differentiate f  x   3x cos  2 x 
Using the Product and Chain Rules.
g  x   3x
2
g '  x   6x
h  x   cos  2 x    cos 2 x 
2
2
h '  x   2cos  2 x    sin  2 x    2 
h '  x   4  cos  2 x  sin  2 x  
f '  x   cos 2 (2 x)[6 x]  3x 2  4  cos  2 x  sin  2 x  
f '  x   6 x cos 2 (2 x)  12 x 2 cos  2 x  sin  2 x 
17
2
2
EXAMPLE 10: Differentiate f  x   3x cos  2 x 
Using the Product and Chain Rules.
g  x   3x
2
g '  x   6x
Alternate approach
h  x   cos  2 x    cos 2 x 
2
2
h '  x   2cos  2 x    sin  2 x    2 
h '  x   4  cos  2 x  sin  2 x  
Using the sine double angle identity sin 2A = 2sin A cos A
h '  x   2  2cos  2 x  sin  2 x    2sin  4 x 
f '  x   3x 2  2sin 4 x   cos2  2 x  6 x 
f '  x   6 x2 sin  4 x   6 x cos 2  2 x 
18
Assignment Questions
Differentiate each of the following trigonometric functions.
1 a) y = 4cos x
dy
 4sin x
dx
1 b) f(x) = 3sin(5x2)
f '  x   3cos  5x 2  10 x 
f '  x   30 x cos  5x 2 
sin 2 x
1 c) y 
cos2 x 
2
cos
2
x
2sin
x
cos
x

sin
x  2sin 2 x 


dy

dx
cos 2  2 x 
19
Assignment Questions
Differentiate each of the following trigonometric functions.
d) f(x) = –
2cos3(7x)
 2  cos  7x  
3
REWRITE
f '  x   6  cos  7 x     sin  7 x  7  
2
f '  x   42cos2  7 x  sin  7 x 
e) y = x sin x
y '  x  cos x   sin x 1
y '  x cos x  sin x
20
Assignment Questions
2. Find the slope of the tangent to the function
y = cos2(2x) at the point
x

2
.
dy
 2  cos  2 x    2    sin  2 x  
dx
dy
 4cos  2 x  sin  2 x 
dx
The slope of the tangent at
x
dy
 2  2cos  2 x  sin  2 x  
dx
dy
 2  2sin  2 x  cos  2 x    2sin  4 x 
dx
dy
   
 2sin  4     2sin  2   2  0   0
dx
  2 

2
is 0
21