Download 3.3 - Derivatives of Trigonometric Functions

3.3 - Derivatives of Trigonometric
Functions
1
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.
d
d
2
 sin x   cos x
 tan x   sec x
dx
dx
d
 cos x    sin x
dx
2
Try These
Find the derivative.
(a) f ( x)  3csc x  2cos x
(b) f ( x)  e tan x
x
1  sin t
(c) g (t ) 
t  cos t
3
Examples
1. Find the equation of the tangent line to
the curve at the point.
1
f ( x) 
;  0,1
sin x  cos x
2. Determine
f ( x)  e cos x
x
4
Limit Definitions
y  tan  
y 
 
Since,   tan  for   0, 
 2
sin 

  cos  sin 
cos 
sin 
cos  
1

lim cos   lim
 0
sin 
 lim1

sin 
1  lim
1
 0 
 0
Therefore,
 0
lim
 0
sin 

1
5
Limit Definitions
lim
 0
cos  1

lim
 0
cos   1

0
 cos  1 cos  1 
 lim 


 0
cos  1 
 
cos 2   1
 sin 2 
 lim
 lim
 0   cos   1
 0   cos   1
sin 
sin 
 0 
  lim
 lim
 1  
0

 0
  0  cos   1
 11 
6
Try These
In both of the previous definitions, θ can take
on many forms. Here are a few examples.
sin  3x 
lim
1
x 0  3 x 
sin
lim
x 0


2x
2x

 1
1 
cos  y   1
3 

lim
0
y 0
1 
 y
3 
7
Try These
Evaluate.
sin 4 x
(a ) lim
x0 sin 6 x
A common effective
strategy is to separate the
quotient into a product.
(c) lim
t 0
cos   1
(b) lim
 0 sin 
sin 2  3t 
t
2
sin x  cos x
(d ) lim
x0
cos 2 x
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