AP Calculus AB - Worksheet 25
Derivatives of sine and cosine functions.
Know the following theorems:
d
sin
dx
 cos

d
dx
d
cos
dx
and
Examples
  sin
1. y  sin5x
2. y  cos9x
y  sin 5x
y  cos 9x
y '  cos 5x  d 5x
y '   sin 9x  d 9x

d
dx
3. y  3sin 4 x
4
y  3 sin x
3
y '  3 4 sin x
y '  9sin 9x
ÊÊy '  5cos 5x
 d sin x
3
y '  12 sin x
 cos x  d x
ÊÊy '  12sin 3 x cos x
Find the derivative of each function. Simplify, if necessary.
1
y  sin 3x
2
y  5 cos 2x
3
f x   cos2 x
4
5
6
7
8
y  sin x cos x
y  2x cos x
1
f x    5 sin x
x
4
y
cos x
cos x
f x  
1 sin x
9
y  1 cos 2x 
10
Find the equations for the lines that are tangent and normal to the graph of f x   sin x  3 at x   .
11
Find the equation of the normal line to f x   sin x  cos x at x   .
12
Find the derivative of y  cos2 x 3  4x
13
Determine all values of x in the interval 0, 2  for which f x   cos 2x has horizontal tangents.
2

Answers
1) y '  3cos3x
4) y'  cos 2x (common identity)

3) f ' x   2 cos x sin x
dy
 10 sin 2x
dx
5) y'  2 cos x  x sin x 
2)
1
 5 cos x
x2
10) T : y  3  1x   
6) f ' x   
7) y'  4 sec x tan x
8) f ' x   
11) y 1  x  
12) y'  2 cos x  4x sin x  4x 3x  4
1
1 sin x

3
 
3

2

N : y  3  1x   