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3.3 - Derivatives of Trigonometric Functions

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Derivative Definitions

We can now use the limit of the difference quotient and the sum/difference formulas for trigonometric functions to determine the following derivatives.

sin cosd

x xdx

cos sind

x xdx

2tan secd

x xdx

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Try These

( ) ( ) 3csc 2cosa f x x x

( ) ( ) tanxb f x e x

1 sin( ) ( )

cos

tc g t

t t

Find the derivative.

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Examples

1( ) ; 0,1

sin cosf x

x x

1. Find the equation of the tangent line to the curve at the point.

( ) cosxf x e x

2. Determine

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Limit Definitions

Since, tan 0,2

for

Therefore,

y

tany

sin

cos

sincos 1

0

sin1 lim 1

0 0 0

sinlim cos lim lim1

cos sin

0

sinlim 1

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Limit Definitions

0 0

cos 1 cos 1 cos 1lim lim

cos 1

2 2

0 0

cos 1 sinlim lim

cos 1 cos 1

0 0

sin sin 0lim lim 1 0

cos 1 1 1

0

cos 1lim 0

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Try These

In both of the previous definitions, θ can take on many forms. Here are a few examples.

1

3

3sinlim

0

x

xx

0

sin 2lim 1

2x

x

x

0

1cos 1

3lim 0

13

y

y

y

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Try These

0

sin 4( ) lim

sin 6x

xa

x

0

cos 1( ) lim

sinb

2

20

sin 3( ) lim

t

tc

t

0

sin cos( ) lim

cos 2x

x xd

x

Evaluate. A common effective strategy is to separate the quotient into a product.