Derivatives of the Trigonometric Functions
Marius Ionescu
10/01/2010
Marius Ionescu
Derivatives of the Trigonometric Functions
The Derivative of sin
d sin x
dx
=
lim
sin(x + h) − sin x
h→0
h
Marius Ionescu
Derivatives of the Trigonometric Functions
The Derivative of sin
d sin x
dx
=
=
lim
h→0
lim
sin(x + h) − sin x
h
sin x cos h + cos x sin h − sin x
h→0
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h
Derivatives of the Trigonometric Functions
The Derivative of sin
d sin x
dx
=
=
lim
h→0
lim
sin(x + h) − sin x
h
sin x cos h + cos x sin h − sin x
h→0
= sin x lim
h →0
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cos h − 1
h
h
+ cos x lim
h →0
sin h
h
Derivatives of the Trigonometric Functions
Useful limits
Theorem
lim
and
sin θ
θ→0
lim
θ
= 1,
cos θ − 1
θ→0
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θ
= 0.
Derivatives of the Trigonometric Functions
Example
Example
Evaluate
lim
x →0
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sin 3x
x
Derivatives of the Trigonometric Functions
Theorem
Theorem
The derivatives of sin x and cos x are given by
d
dx sin x = cos x
d cos x = − sin x .
dx
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Derivatives of the Trigonometric Functions
Examples
Example
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Derivatives of the Trigonometric Functions
Examples
Example
Dierentiate sin(2x ).
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Derivatives of the Trigonometric Functions
Examples
Example
Dierentiate sin(2x ).
Dierentiate f (x ) = sin
x 2 + x1 .
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Derivatives of the Trigonometric Functions
Examples
Example
Dierentiate sin(2x ).
Dierentiate f (x ) = sin x 2 + x1 .
√
Dierentiate y = cos(3x + x ).
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Derivatives of the Trigonometric Functions
More Examples
Example
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Derivatives of the Trigonometric Functions
More Examples
Example
Dierentiate
y
= sin x cos x
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Derivatives of the Trigonometric Functions
More Examples
Example
Dierentiate
y
Dierentiate
= sin x cos x
y = sin2 (cos(x 2 + 2))
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Derivatives of the Trigonometric Functions
Other trigonometric functions
Theorem
Derivatives of other trigonometric functions
d tan x
dx
d
dx cot x
d sec x
dx
d csc x
dx
= sec2 x
= − csc2 x
= sec x tan x
= − csc x cot x
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Derivatives of the Trigonometric Functions
Example
Example
Find
d sec2 x
dx
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Derivatives of the Trigonometric Functions
Example
Example
Compute the derivative of
y = (sin3 (tan2 (2x )))4 .
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Derivatives of the Trigonometric Functions