Derivatives of Trigonometric Functions

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Lesson 3-4

• Derivatives of Trigonometric Functions

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Objectives

• Find the derivatives of trigonometric functions

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Vocabulary

• none new

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Trig Differentiation Rules
d ---- (sin x) cos x Sin dx d ---- ( cos x)
-sin x Cos dx Rest of them can be done from
these two and the Quotient rule!
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Rest of Trig Differentiation Rules
d ---- tan x sec² x dx d ---- (cot x)
-csc² x dx d ---- sec x (sec x) (tan
x) dx d ---- (csc x) -(csc x) (cot x) dx
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Example 1
Find the derivatives of the following

1. y sin x cos x
2. f(x) (tan x) / (x 1)

y(x) cos x (-sin x) cos x sin x
(x 1) (sec² x) (1) (tan x) y(x)
---------------------------------------
(x 1)²
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Example 2
Find the derivatives of the following

1. y sin (p/4)
2. y x³ sin x

y(x) 0
y(x) 3x² (sin x) x³ (cos x)
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Example 3
Find the derivatives of the following

1. y x² 2x cos x
2. y x / (sec x 1)

y(x) 2x 2x (-sin x) (2) (cos x)
(sec x 1) (1) (sec x tan x)
(x) y(x) --------------------------------------
-------- (sec x
1)²
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Example 4
Find the derivatives of the following

1. y x / (cot x)
2. y (csc x) / ex

(cot x) (1) (-csc² x) (x) y(x)
---------------------------------------
(cot x)²
(ex) (-csc x cot x) (ex) (csc
x) y(x) ---------------------------------------
------ (ex)²
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Example 5
Find the derivatives of the following

• sin x
• lim ----------- 7x
  sin 5x
• lim ----------- 5x

x?0
x?0
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Example 6
A particle moves along a line so that at any time
tgt0 its position is given by x(t) 2pt
cos(2pt).   Find the velocity at time
t.     Find the speed (v) at t ½.     What
are the values of t for which the particle is at
rest?
v(t) x(t) 2p 2p (sin 2pt)
Speed v(1/2) x(t) 2p 2p (sin p)
2p
When v(t) 0 2p 2p (sin 2pt) 2p (sin 2pt)
2p ? sin 2pt 1
? t ¼, 5/4, 9/4, 13/4,
etc
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Summary Homework

• Summary
• Use trig rules for finding derivatives
• d(sin x) cos x
• d(cos x) -sin x
• Homework
• pg 216 217 1-3, 6, 9-11, 18, 29, 41