Transcript Derivatives Involving Trigonometric Functions

Examples A: Derivatives Involving
Algebraic Functions
The derivative of composite function
for the case f(x) = gn(x)
Let:
f(x) = gn(x)
Then:
f' (x) = ngn-1(x) . g'(x)
Example:
Let f(x) = (3x8 - 5x + 3 )20
Then f(x) = 20 (3x8 - 5x + 3 )19 (24x7 - 5)
Examples (1)
Find the derivativeof each of the following functions:
1.
2.
1 9
1
1
1
5
9
9
5
f ( x)  x  9  x  x  x  9 

x
x 9 x5 5 x9
9
 9
3 
f ( x)  2 x 

9
5
x 

7
7
3.
 9
3   6 4
f ( x)  2 x 
5x  


9
5
x
x

 
3
4.
5.
f ( x) 
4
f ( x) 
4
2 x
8
2 x

5
 x  1 4.
2
8
f ( x) 

 x 1
5
6.
3x  x  
f ( x)  9
x  3x  7
7.
f ( x)  x ( x  1)  3x  x
4
2
9
7
3
4
2 x
8

 x 1
5
1 9
1 9 5
1. f ( x)  x  9  x  9  x
x
x
1
1
9
5
 x 

5
9
9
5
x
x
9
9
1
9
 19
5
9
9
5
 95
x x x x x x x x
9
 95

f ( x)  9 x  9 x
8
10
1  89 1  109 5  94
 x  x  x
9
9
9
9 54 5  149 9  145
 x  x  x
5
9
5
2.

 9
3 
f ( x)  2 x  5 
9
x 

 2 x  3x

9

7

 95 7
f ( x)  7 2 x  3x
9

 95 6
27  145 

8
 18x 
x 
5


7
3.

 9
3   6 4
f ( x)  2 x  5
5x  


9
x
x

 
 2 x  3x

9

 95
 5x  4x
7

 5x  4 x
6

1 3
  35x  4x  30x  4x
27


  72 x  3x  18x  5 x 


f ( x)  2 x  3 x
9
6
3
1 3
 95
7
9
1 2
6
 95
6
8
5
 145
2

2 x
 2 x  x  1
4.
f ( x) 
4
8

 x 1
5
5
4
8


 16x
5
8
f ( x)  2 x  x  1
4
1
4
7

1
5.
2
f ( x) 
2 x
 22 x  x  1
4
8

 x 1
5
 54
8


 16x
5 8
f ( x)   2 x  x  1
2
 94
7

1
6.
3x 4  x 2  
f ( x)  9
x  3x  7

f ( x) 
( x 9  3 x  7)(12x 3  2 x)  (3 x 4  x 2   )(9 x 8  3)
x
9
 3x  7

2
7.
f ( x)  x ( x  1)  3 x  3 x
9
7
1 1
3 2
 [ x ( x  1)  3 x  x ]
9
7

1
1

3
2
1 9
7
f ( x)  [ x ( x  1)  3 x  x ] 
2
2

1 3
9
6
7
8
[ x  7( x  1)  ( x  1)  9 x  3  x ]
3
Example (2)
Let : f ( x)  g (h( x))
g (1)  4 & h(2)  1, h(2)  7
Find f (2)
Soluion :
f ( x)  g (h( x))
 f ( x)  g (h( x)) h( x)
 f (2)  g (h(2)) h(2)
 f (2)  g (1) h(2)
 g (1)) h(2)  4(7)  28
Examples B: Derivatives Involving
Trigonometric Functions
Basic Formulas
1. y  sin x  y  cos x
2. y  cos x  y   sin x
3. y  tan x  y  sec x
2
4. y  cot x  y   csc x
5. y  sec x  y  sec x tan x
5. y  csc x  y   csc x cot x
2
General Formulas (Chain Rule)
Let u=u(x)
1. y  sin u  y  cos u  u 
2. y  cos u  y   sin u  u 
3. y  tan u  y  sec u  u 
2
4. y  cot u  y   csc u  u 
5. y  sec u  y  sec u tan u  u 
5. y  csc u  y   csc u cot u  u 
2
Examples (1)
1. y  x 8 sin 2 x
 y  x 8 cos 2 x  2  sin 2 x  8 x 7
 2 x 8 cos 2 x  8 x 7 sin 2 x
tan x
x  x sec x
( x  x sec x) sec 2 x  tan x(1  x sec x tan x  sec x)
 y 
( x  x sec x) 2
2. y 
3. y  cos x cot x
 y  cos x( csc 2 x)  cot x( sin x)
  cos x csc 2 x  cot x sin x
cos x
sin x
  cos x(csc 2 x  1) (not much improvement!)
  cos x csc 2 x  cos x
; because cot x 
Examples (2)
1. y  cos x
20
20
19

 y   sin x  20x
2. y  t an x 20
 y   sec 2 x 20  20x19
3. y  sec x
20
 y   sec x 20 t an x 20  20x19
Examples (3)
1. y  cos 20 x  (cos x) 20
 y  20 cos19 x  ( sin x)
2. y  tan 20 x  (tan x) 20
 y  20 tan19 x  sec 2 x
3. y  sec 20 x  (sec x) 20
 y  20 sec19 x  sec x tan x
4. y  sin 20 x  (sin x) 20
 y  20 sin19 x  cos x
Examples (4)
1. y  x 8 sin 5 x
( product!)
 y  x 8  cos 5 x  5  sin 5 x  8 x 7
 5 x 8 cos 5 x  8 x 7 sin 5 x
5 sin x
(quotient!)
1
cot x   6
x
1


cot
x

 6   5 cos x  5 sin x  csc 2 x  x  2

x

 y  
2
1


 cot x   6 
x



1
1


Note :  x 1 and so     x  2
x
 x
2. y 


3. y  (3 csc x  x  3  )100 ( Power!)
1
1 2
99
3
 y  100 (3 csc x  x   )  (3 csc x cot x  x )
2
1
1


1
Note : x  x 2 and so x  x 2
2
 
4. y  (2x  3  43 tan 2 x 5 )10
2
3
 (2x  3  4 tan x 5 )10
2
5 3 10
 [2x  3  4(tan x ) ]
2
5 3 9
 y  10[2x  3  4(tan x ) ] 
1
5 3
2
[2  4  (tan x )  sec 2 x 5  5 x 4 ]
3
2
5 3 9
 y  10[2x  3  4(tan x ) ]
40 x 4 sec 2 x 5
[2 
]
3 3 tan x 5
Examples (6)
1. y  csc 5 (2 x 4  x  3) 7
 csc(2 x 4  x  3)
7
5
7
5
7
5
2
7
 y   csc(2 x 4  x  3)  cot(2 x 4  x  3)  (2 x 4  x  3) 5  (8 x 3  1)
5
2. y  5 csc 7 (2 x 4  x  3)
7
5
 csc (2 x  x  3)  [csc(2 x  x  3)]
4
4
2
7
5
7
 y  [csc(2 x 4  x  3)] 5  [ csc(2 x 4  x  3) cot(2 x 4  x  3)]  (8 x 3  1)
5
Examples (7)
1. y  sin(tan 8 x )
5
2. y  sin (tan 8 x )
9
5
3. y  sin (tan(8 x  1) )
9
5
3
1. y  sin(tan 8 x )
5
y  cos(tan 8 x )
5
 sec 8 x
2
 40 x
4
5
2. y  sin (tan 8 x )
9
5
 [sin(tan 8 x )]
5
9
y  9[sin(tan 8 x )]
5
 cos(tan 8 x )
5
 sec 8 x
2
 40 x
4
5
8
3. y  sin [ tan(8 x  1) ]
9

5
7

 sin[ tan(8 x  1) ]
5
7

9

y  9 sin[ tan(8 x  1) ]
5
 cos[ tan(8 x  1) ]
5
 sec (8 x  1)
2
5
 7(8 x  1)
5
 40 x
4
6
7
7
7
8