Derivatives of the Trigonometric Functions

AP C ALCULUS
D ERIVATIVES OF THE T RIGONOMETRIC F UNCTIONS
1. lim (x2 + cos x) = 1
x→0
−1 +
2. lim (sin x − cos x) =
2
x→π/3
√
2 2
sin 5t
=−
3. lim
t
π
t→π/4
√
3
sin(cos x)
= sin 1
sec x
√
sin x
2 2
=
5. lim
3π
x→π/4 3x
4. lim
x→0
6.
lim
x→π/4
1
tan x
=
4x
π
sin2 x
sin x
= lim
sin x = 0
x→0 x
x→0
x
7. lim
8.
tan 3x
lim
x→0 3 tan 2x
=
=
9.
10.
11.
12.
sin 3x
cos
3x
lim
x→0 3 sin 2x
cos 2x
lim
x→0
sin 3x cos 2x
cos 3x 3 sin 2x
=
sin 3x
3x cos 2x
lim 3x
3 sin 2x
x→0 cos 3x
2x
2x
=
1
2
dy
= − sin x − 2 sec2 x
dx
dy
= cos x − sin x
dx
dy
dx
dy
dx
=
(x)(− csc x cot x) + (csc x)(1)
=
csc x − x csc x cot x
=
(csc x)(− csc2 x) + (cot x)(− csc x cot x)
=
− csc3 x − cot2 x csc x
=
− csc x(csc2 x + cot2 x)
1
13.
14.
15.
16.
dy
dx
=
(1 + cos x)(cos x) − (sin x)(− sin x)
(1 + cos x)2
=
cos x + cos2 x + sin2 x
(1 + cos x)2
=
1 + cos x
(1 + cos x)2
=
1
1 + cos x
dy
x sec2 x − tan x
=
dx
x2
dy
dx
dy
dx
=
(sin x + cos x)(1) − x(cos x − sin x)
(sin x + cos x)2
=
sin x + cos x − x cos x + x sin x
(sin x + cos x)2
=
(x−3 )(sin x sec2 x + tan x cos x) + (sin x tan x)(−3x−4 )
=
sin x sec2 x + tan x cos x −3 sin x tan x
+
x3
x4
x sin x sec2 x + x tan x cos x − 3 sin x tan x
x4
£ 2
¤
(sec x) (x )(sec2 x) + (tan x)(2x) − (x2 tan x)(sec x tan x)
sec2 x
=
17.
dy
dx
=
x2 sec3 x + 2x sec x tan x − x2 sec x tan2 x
sec2 x
=
18. Slope of tangent
¯
dy ¯¯
dy
= x(− sin x) + cos x −→
= −1 −→ mT = −1
dx
dx ¯x=π
Point
Given as (π, −π)
Equation of tangent
y + π = −(x − π)
19. Slope of tangent
¯
dy ¯¯
dy
= sec2 x −→
= 2 −→ mT = 2
dx
dx ¯x=π/4
Point
Given as
³π
4
´
,1
Equation of tangent
³
π´
y−1=2 x−
4
2
20. Slope of tangent
g 0 (x) = 2 cos x −→ g 0
³π ´
6
=
√
3 −→ mT =
√
3
Point
³π´
³π ´
g
= 1 −→
,1
6
6
Equation of tangent
√ ³
π´
y−1= 3 x−
6
21. For f to have a horizontal tangent, f 0 (x) = 0.
f 0 (x) = 1 + 2 cos x
f 0 (x) = 0 when 1 + 2 cos x = 0 −→ x =
4π
2π
+ 2kπ or x =
+ 2kπ
3
3
3