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The Derivatives of the Trigonometric Functions Bernd Schr oder - - PowerPoint PPT Presentation

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Two Limits Derivatives Examples The Derivatives of the Trigonometric Functions Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions Two

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SLIDE 1

logo1 Two Limits Derivatives Examples

The Derivatives of the Trigonometric Functions

Bernd Schr¨

  • der

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-2
SLIDE 2

logo1 Two Limits Derivatives Examples

sin(θ) ≤ θ ≤ tan(θ)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-3
SLIDE 3

logo1 Two Limits Derivatives Examples

sin(θ) ≤ θ ≤ tan(θ)

✲

x

✻

y Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

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SLIDE 4

logo1 Two Limits Derivatives Examples

sin(θ) ≤ θ ≤ tan(θ)

✲

x

✻

y

✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ✧

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-5
SLIDE 5

logo1 Two Limits Derivatives Examples

sin(θ) ≤ θ ≤ tan(θ)

✲

x

✻

y

✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ✧ ❑

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-6
SLIDE 6

logo1 Two Limits Derivatives Examples

sin(θ) ≤ θ ≤ tan(θ)

✲

x

✻

y

✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ✧ ❑

θ Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-7
SLIDE 7

logo1 Two Limits Derivatives Examples

sin(θ) ≤ θ ≤ tan(θ)

✲

x

✻

y

✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ✧ ❑

θ Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-8
SLIDE 8

logo1 Two Limits Derivatives Examples

sin(θ) ≤ θ ≤ tan(θ)

✲

x

✻

y

✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ✧ ❑

θ 1 Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-9
SLIDE 9

logo1 Two Limits Derivatives Examples

sin(θ) ≤ θ ≤ tan(θ)

✲

x

✻

y

✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ✧ ❑

θ 1

r

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-10
SLIDE 10

logo1 Two Limits Derivatives Examples

sin(θ) ≤ θ ≤ tan(θ)

✲

x

✻

y

✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ✧ ❑

θ 1

r

(x,y) Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

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SLIDE 11

logo1 Two Limits Derivatives Examples

sin(θ) ≤ θ ≤ tan(θ)

✲

x

✻

y

✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ✧ ❑

θ 1

r

(x,y) θ Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

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SLIDE 12

logo1 Two Limits Derivatives Examples

sin(θ) ≤ θ ≤ tan(θ)

✲

x

✻

y

✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ✧ ❑

θ 1

r

(x,y) θ Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

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SLIDE 13

logo1 Two Limits Derivatives Examples

sin(θ) ≤ θ ≤ tan(θ)

✲

x

✻

y

✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ✧ ❑

θ 1

r

(x,y) θ sin(θ) Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

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SLIDE 14

logo1 Two Limits Derivatives Examples

sin(θ) ≤ θ ≤ tan(θ)

✲

x

✻

y

✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ✧ ❑

θ 1

r

(x,y) θ sin(θ)

sin(θ) ≤ θ

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

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SLIDE 15

logo1 Two Limits Derivatives Examples

sin(θ) ≤ θ ≤ tan(θ)

✲

x

✻

y

✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ✧ ❑

θ 1

r

(x,y) θ sin(θ)

sin(θ) ≤ θ

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

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SLIDE 16

logo1 Two Limits Derivatives Examples

sin(θ) ≤ θ ≤ tan(θ)

✲

x

✻

y

✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ✧ ❑

θ 1

r

(x,y) θ sin(θ)

sin(θ) ≤ θ

tan(θ) Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

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SLIDE 17

logo1 Two Limits Derivatives Examples

sin(θ) ≤ θ ≤ tan(θ)

✲

x

✻

y

✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ✧ ❑

θ 1

r

(x,y) θ sin(θ)

sin(θ) ≤ θ

tan(θ) Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

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SLIDE 18

logo1 Two Limits Derivatives Examples

sin(θ) ≤ θ ≤ tan(θ)

✲

x

✻

y

✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ✧ ❑

θ 1

r

(x,y) θ sin(θ)

sin(θ) ≤ θ

tan(θ)

q

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

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SLIDE 19

logo1 Two Limits Derivatives Examples

sin(θ) ≤ θ ≤ tan(θ)

✲

x

✻

y

✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ✧ ❑

θ 1

r

(x,y) θ sin(θ)

sin(θ) ≤ θ

tan(θ)

q P

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

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SLIDE 20

logo1 Two Limits Derivatives Examples

sin(θ) ≤ θ ≤ tan(θ)

✲

x

✻

y

✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ✧ ❑

θ 1

r

(x,y) θ sin(θ)

sin(θ) ≤ θ

tan(θ)

q P θ ≤ tan(θ)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

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SLIDE 21

logo1 Two Limits Derivatives Examples

sin(θ) ≤ θ ≤ tan(θ)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

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SLIDE 22

logo1 Two Limits Derivatives Examples

sin(θ) ≤ θ ≤ tan(θ)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

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SLIDE 23

logo1 Two Limits Derivatives Examples

sin(θ) ≤ θ ≤ tan(θ)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

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SLIDE 24

logo1 Two Limits Derivatives Examples

sin(θ) ≤ θ ≤ tan(θ)

✲

x

✻

y

✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ✧ ❑

θ 1

r

(x,y) θ sin(θ)

sin(θ) ≤ θ

tan(θ)

q P θ ≤ tan(θ)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

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SLIDE 25

logo1 Two Limits Derivatives Examples

sin(θ) ≤ θ ≤ tan(θ)

✲

x

✻

y

✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ✧ ❑

θ 1

r

(x,y) θ sin(θ)

sin(θ) ≤ θ

tan(θ)

q P θ ≤ tan(θ)

1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

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SLIDE 26

logo1 Two Limits Derivatives Examples

sin(θ) ≤ θ ≤ tan(θ)

✲

x

✻

y

✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ✧ ❑

θ 1

r

(x,y) θ sin(θ)

sin(θ) ≤ θ

tan(θ)

q P θ ≤ tan(θ)

1 ≥ sin(θ) θ

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

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SLIDE 27

logo1 Two Limits Derivatives Examples

sin(θ) ≤ θ ≤ tan(θ)

✲

x

✻

y

✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ✧ ❑

θ 1

r

(x,y) θ sin(θ)

sin(θ) ≤ θ

tan(θ)

q P θ ≤ tan(θ)

1 ≥ sin(θ) θ = y θ

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

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SLIDE 28

logo1 Two Limits Derivatives Examples

sin(θ) ≤ θ ≤ tan(θ)

✲

x

✻

y

✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ✧ ❑

θ 1

r

(x,y) θ sin(θ)

sin(θ) ≤ θ

tan(θ)

q P θ ≤ tan(θ)

1 ≥ sin(θ) θ = y θ ≥ y tan(θ)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-29
SLIDE 29

logo1 Two Limits Derivatives Examples

sin(θ) ≤ θ ≤ tan(θ)

✲

x

✻

y

✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ✧ ❑

θ 1

r

(x,y) θ sin(θ)

sin(θ) ≤ θ

tan(θ)

q P θ ≤ tan(θ)

1 ≥ sin(θ) θ = y θ ≥ y tan(θ) = y y/x

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-30
SLIDE 30

logo1 Two Limits Derivatives Examples

sin(θ) ≤ θ ≤ tan(θ)

✲

x

✻

y

✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ✧ ❑

θ 1

r

(x,y) θ sin(θ)

sin(θ) ≤ θ

tan(θ)

q P θ ≤ tan(θ)

1 ≥ sin(θ) θ = y θ ≥ y tan(θ) = y y/x = x

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-31
SLIDE 31

logo1 Two Limits Derivatives Examples

sin(θ) ≤ θ ≤ tan(θ)

✲

x

✻

y

✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ✧ ❑

θ 1

r

(x,y) θ sin(θ)

sin(θ) ≤ θ

tan(θ)

q P θ ≤ tan(θ)

1 ≥ sin(θ) θ = y θ ≥ y tan(θ) = y y/x = x → 1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-32
SLIDE 32

logo1 Two Limits Derivatives Examples

sin(θ) ≤ θ ≤ tan(θ)

✲

x

✻

y

✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ✧ ❑

θ 1

r

(x,y) θ sin(θ)

sin(θ) ≤ θ

tan(θ)

q P θ ≤ tan(θ)

1 ≥ sin(θ) θ = y θ ≥ y tan(θ) = y y/x = x → 1 (θ → 0)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

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SLIDE 33

logo1 Two Limits Derivatives Examples Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

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SLIDE 34

logo1 Two Limits Derivatives Examples

So lim

θ→0+

sin(θ) θ = 1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-35
SLIDE 35

logo1 Two Limits Derivatives Examples

So lim

θ→0+

sin(θ) θ = 1 and

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-36
SLIDE 36

logo1 Two Limits Derivatives Examples

So lim

θ→0+

sin(θ) θ = 1 and, because sin(θ) θ is an even function

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-37
SLIDE 37

logo1 Two Limits Derivatives Examples

So lim

θ→0+

sin(θ) θ = 1 and, because sin(θ) θ is an even function, lim

θ→0−

sin(θ) θ = 1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-38
SLIDE 38

logo1 Two Limits Derivatives Examples

So lim

θ→0+

sin(θ) θ = 1 and, because sin(θ) θ is an even function, lim

θ→0−

sin(θ) θ = 1, that is lim

θ→0

sin(θ) θ = 1.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-39
SLIDE 39

logo1 Two Limits Derivatives Examples

So lim

θ→0+

sin(θ) θ = 1 and, because sin(θ) θ is an even function, lim

θ→0−

sin(θ) θ = 1, that is lim

θ→0

sin(θ) θ = 1. Now lim

θ→0

1−cos(θ) θ

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-40
SLIDE 40

logo1 Two Limits Derivatives Examples

So lim

θ→0+

sin(θ) θ = 1 and, because sin(θ) θ is an even function, lim

θ→0−

sin(θ) θ = 1, that is lim

θ→0

sin(θ) θ = 1. Now lim

θ→0

1−cos(θ) θ 1+cos(θ) 1+cos(θ)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-41
SLIDE 41

logo1 Two Limits Derivatives Examples

So lim

θ→0+

sin(θ) θ = 1 and, because sin(θ) θ is an even function, lim

θ→0−

sin(θ) θ = 1, that is lim

θ→0

sin(θ) θ = 1. Now lim

θ→0

1−cos(θ) θ 1+cos(θ) 1+cos(θ) = lim

θ→0

1−cos2(θ) θ(1+cos(θ))

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-42
SLIDE 42

logo1 Two Limits Derivatives Examples

So lim

θ→0+

sin(θ) θ = 1 and, because sin(θ) θ is an even function, lim

θ→0−

sin(θ) θ = 1, that is lim

θ→0

sin(θ) θ = 1. Now lim

θ→0

1−cos(θ) θ 1+cos(θ) 1+cos(θ) = lim

θ→0

1−cos2(θ) θ(1+cos(θ)) = lim

θ→0

sin2(θ) θ(1+cos(θ))

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-43
SLIDE 43

logo1 Two Limits Derivatives Examples

So lim

θ→0+

sin(θ) θ = 1 and, because sin(θ) θ is an even function, lim

θ→0−

sin(θ) θ = 1, that is lim

θ→0

sin(θ) θ = 1. Now lim

θ→0

1−cos(θ) θ 1+cos(θ) 1+cos(θ) = lim

θ→0

1−cos2(θ) θ(1+cos(θ)) = lim

θ→0

sin2(θ) θ(1+cos(θ)) = lim

θ→0sin(θ)sin(θ)

θ 1 1+cos(θ)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-44
SLIDE 44

logo1 Two Limits Derivatives Examples

So lim

θ→0+

sin(θ) θ = 1 and, because sin(θ) θ is an even function, lim

θ→0−

sin(θ) θ = 1, that is lim

θ→0

sin(θ) θ = 1. Now lim

θ→0

1−cos(θ) θ 1+cos(θ) 1+cos(θ) = lim

θ→0

1−cos2(θ) θ(1+cos(θ)) = lim

θ→0

sin2(θ) θ(1+cos(θ)) = lim

θ→0sin(θ)sin(θ)

θ 1 1+cos(θ) = 0·1· 1 2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-45
SLIDE 45

logo1 Two Limits Derivatives Examples

So lim

θ→0+

sin(θ) θ = 1 and, because sin(θ) θ is an even function, lim

θ→0−

sin(θ) θ = 1, that is lim

θ→0

sin(θ) θ = 1. Now lim

θ→0

1−cos(θ) θ 1+cos(θ) 1+cos(θ) = lim

θ→0

1−cos2(θ) θ(1+cos(θ)) = lim

θ→0

sin2(θ) θ(1+cos(θ)) = lim

θ→0sin(θ)sin(θ)

θ 1 1+cos(θ) = 0·1· 1 2 = 0.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-46
SLIDE 46

logo1 Two Limits Derivatives Examples

lim

θ→0

sin(θ) θ = 1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-47
SLIDE 47

logo1 Two Limits Derivatives Examples

lim

θ→0

sin(θ) θ = 1

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-48
SLIDE 48

logo1 Two Limits Derivatives Examples

lim

θ→0

1−cos(θ) θ = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-49
SLIDE 49

logo1 Two Limits Derivatives Examples

lim

θ→0

1−cos(θ) θ = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-50
SLIDE 50

logo1 Two Limits Derivatives Examples

The Derivative of the Sine Function

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-51
SLIDE 51

logo1 Two Limits Derivatives Examples

The Derivative of the Sine Function

d dx sin(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-52
SLIDE 52

logo1 Two Limits Derivatives Examples

The Derivative of the Sine Function

d dx sin(x) = lim

h→0

sin(x+h)−sin(x) h

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-53
SLIDE 53

logo1 Two Limits Derivatives Examples

The Derivative of the Sine Function

d dx sin(x) = lim

h→0

sin(x+h)−sin(x) h = lim

h→0

sin(x)cos(h)+cos(x)sin(h)−sin(x) h

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-54
SLIDE 54

logo1 Two Limits Derivatives Examples

The Derivative of the Sine Function

d dx sin(x) = lim

h→0

sin(x+h)−sin(x) h = lim

h→0

sin(x)cos(h)+cos(x)sin(h)−sin(x) h = lim

h→0

cos(x)sin(h) h

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-55
SLIDE 55

logo1 Two Limits Derivatives Examples

The Derivative of the Sine Function

d dx sin(x) = lim

h→0

sin(x+h)−sin(x) h = lim

h→0

sin(x)cos(h)+cos(x)sin(h)−sin(x) h = lim

h→0

cos(x)sin(h) h + lim

h→0

sin(x)cos(h)−sin(x) h

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-56
SLIDE 56

logo1 Two Limits Derivatives Examples

The Derivative of the Sine Function

d dx sin(x) = lim

h→0

sin(x+h)−sin(x) h = lim

h→0

sin(x)cos(h)+cos(x)sin(h)−sin(x) h = lim

h→0

cos(x)sin(h) h + lim

h→0

sin(x)cos(h)−sin(x) h = cos(x) lim

h→0

sin(h) h

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-57
SLIDE 57

logo1 Two Limits Derivatives Examples

The Derivative of the Sine Function

d dx sin(x) = lim

h→0

sin(x+h)−sin(x) h = lim

h→0

sin(x)cos(h)+cos(x)sin(h)−sin(x) h = lim

h→0

cos(x)sin(h) h + lim

h→0

sin(x)cos(h)−sin(x) h = cos(x) lim

h→0

sin(h) h +sin(x) lim

h→0

cos(h)−1 h

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-58
SLIDE 58

logo1 Two Limits Derivatives Examples

The Derivative of the Sine Function

d dx sin(x) = lim

h→0

sin(x+h)−sin(x) h = lim

h→0

sin(x)cos(h)+cos(x)sin(h)−sin(x) h = lim

h→0

cos(x)sin(h) h + lim

h→0

sin(x)cos(h)−sin(x) h = cos(x) lim

h→0

sin(h) h +sin(x) lim

h→0

cos(h)−1 h = cos(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-59
SLIDE 59

logo1 Two Limits Derivatives Examples

The Derivative of the Cosine Function

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-60
SLIDE 60

logo1 Two Limits Derivatives Examples

The Derivative of the Cosine Function

d dx cos(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-61
SLIDE 61

logo1 Two Limits Derivatives Examples

The Derivative of the Cosine Function

d dx cos(x) = d dx sin π 2 −x

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-62
SLIDE 62

logo1 Two Limits Derivatives Examples

The Derivative of the Cosine Function

d dx cos(x) = d dx sin π 2 −x

  • = cos

π 2 −x

  • ·(−1)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-63
SLIDE 63

logo1 Two Limits Derivatives Examples

The Derivative of the Cosine Function

d dx cos(x) = d dx sin π 2 −x

  • = cos

π 2 −x

  • ·(−1) = −sin(x).

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-64
SLIDE 64

logo1 Two Limits Derivatives Examples

The Derivative of the Tangent Function

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-65
SLIDE 65

logo1 Two Limits Derivatives Examples

The Derivative of the Tangent Function

d dx tan(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-66
SLIDE 66

logo1 Two Limits Derivatives Examples

The Derivative of the Tangent Function

d dx tan(x) = d dx sin(x) cos(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-67
SLIDE 67

logo1 Two Limits Derivatives Examples

The Derivative of the Tangent Function

d dx tan(x) = d dx sin(x) cos(x) = cos(x)cos(x)−

  • −sin(x)
  • sin(x)

cos2(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-68
SLIDE 68

logo1 Two Limits Derivatives Examples

The Derivative of the Tangent Function

d dx tan(x) = d dx sin(x) cos(x) = cos(x)cos(x)−

  • −sin(x)
  • sin(x)

cos2(x) = cos2(x)+sin2(x) cos2(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-69
SLIDE 69

logo1 Two Limits Derivatives Examples

The Derivative of the Tangent Function

d dx tan(x) = d dx sin(x) cos(x) = cos(x)cos(x)−

  • −sin(x)
  • sin(x)

cos2(x) = cos2(x)+sin2(x) cos2(x) = 1 cos2(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-70
SLIDE 70

logo1 Two Limits Derivatives Examples

The Derivatives of the Trigonometric Functions

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-71
SLIDE 71

logo1 Two Limits Derivatives Examples

The Derivatives of the Trigonometric Functions

d dx sin(x) = cos(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-72
SLIDE 72

logo1 Two Limits Derivatives Examples

The Derivatives of the Trigonometric Functions

d dx sin(x) = cos(x) d dx cos(x) = −sin(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-73
SLIDE 73

logo1 Two Limits Derivatives Examples

The Derivatives of the Trigonometric Functions

d dx sin(x) = cos(x) d dx cos(x) = −sin(x) d dx tan(x) = 1 cos2(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-74
SLIDE 74

logo1 Two Limits Derivatives Examples

The Derivatives of the Trigonometric Functions

d dx sin(x) = cos(x) d dx cos(x) = −sin(x) d dx tan(x) = 1 cos2(x)

Secant, cosecant and cotangent are too rarely used to memorize their derivatives.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-75
SLIDE 75

logo1 Two Limits Derivatives Examples

The Derivatives of the Trigonometric Functions

d dx sin(x) = cos(x) d dx cos(x) = −sin(x) d dx tan(x) = 1 cos2(x)

Secant, cosecant and cotangent are too rarely used to memorize their derivatives. (In my opinion.)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-76
SLIDE 76

logo1 Two Limits Derivatives Examples

Compute the Derivative of f(x) = excos(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-77
SLIDE 77

logo1 Two Limits Derivatives Examples

Compute the Derivative of f(x) = excos(x)

f ′(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-78
SLIDE 78

logo1 Two Limits Derivatives Examples

Compute the Derivative of f(x) = excos(x)

f ′(x) = ex cos(x)+

  • −sin(x)
  • ex

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-79
SLIDE 79

logo1 Two Limits Derivatives Examples

Compute the Derivative of f(x) = excos(x)

f ′(x) = ex cos(x)+

  • −sin(x)
  • ex

= ex cos(x)−sin(x)

  • Bernd Schr¨
  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-80
SLIDE 80

logo1 Two Limits Derivatives Examples

Compute the Derivative of f(x) = cos

  • esin(2x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-81
SLIDE 81

logo1 Two Limits Derivatives Examples

Compute the Derivative of f(x) = cos

  • esin(2x)

f ′(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-82
SLIDE 82

logo1 Two Limits Derivatives Examples

Compute the Derivative of f(x) = cos

  • esin(2x)

f ′(x) = −sin

  • esin(2x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-83
SLIDE 83

logo1 Two Limits Derivatives Examples

Compute the Derivative of f(x) = cos

  • esin(2x)

f ′(x) = −sin

  • esin(2x)

esin(2x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-84
SLIDE 84

logo1 Two Limits Derivatives Examples

Compute the Derivative of f(x) = cos

  • esin(2x)

f ′(x) = −sin

  • esin(2x)

esin(2x) cos(2x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-85
SLIDE 85

logo1 Two Limits Derivatives Examples

Compute the Derivative of f(x) = cos

  • esin(2x)

f ′(x) = −sin

  • esin(2x)

esin(2x) cos(2x)2

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-86
SLIDE 86

logo1 Two Limits Derivatives Examples

Compute the Derivative of f(x) = cos

  • esin(2x)

f ′(x) = −sin

  • esin(2x)

esin(2x) cos(2x)2 = −2sin

  • esin(2x)

esin(2x) cos(2x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-87
SLIDE 87

logo1 Two Limits Derivatives Examples

Compute the Derivative of f(x) = sin2(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-88
SLIDE 88

logo1 Two Limits Derivatives Examples

Compute the Derivative of f(x) = sin2(x)

f ′(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-89
SLIDE 89

logo1 Two Limits Derivatives Examples

Compute the Derivative of f(x) = sin2(x)

f ′(x) = 2sin(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions

slide-90
SLIDE 90

logo1 Two Limits Derivatives Examples

Compute the Derivative of f(x) = sin2(x)

f ′(x) = 2sin(x)cos(x)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Derivatives of the Trigonometric Functions