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3.3 DERIVATIVES OF TRIGONOMETRIC FUNCTIONS
For y = f(x), how is
Review
sin2 x + cos2 x =
Find the following limits:
1 !
=
sin x
sin x
=
x"0
x
lim
!
lim
x"0
1# cos x
=
x
!
lim
x"0
1
=
cos x
sin (A + B) =
!
!
d
f (x) defined?
dx
cos (A + B) =
cos x #1
=
x
!
Use the definition of a derivative to find the derivative of f(x) = sin x
d
sin x = cos x
dx
Look at the graph of!
y = sin x
y = cos x
Use the definition of a derivative to find the derivative of f(x) = cos x
d
cos x = "sin x
dx
!
Look at the graph of
y = cos x
y = - sin x
Find the derivative of each of the following:
1.
3.
f (x) = 4 x 2 sin x
4.
g(x) =
f (x) = 3sin x
!
!
2. y = sin x cos x
!
cos x
5x
5.
y = tan x
d
tan x = sec 2 x
dx
!
Use the definition of a derivative to find the derivative of y = cot x
!
d
cot x = "csc 2 x
dx
Use the definition of a derivative to find the derivative of y = sec x
!
d
sec x = sec x " tan x
dx
!
Use the definition of a derivative to find the derivative of y = csc x
Find the derivative of each of the following:
d
csc x = "csc x # cot x
dx
1.
y = sin x – tan x
2.
y = x sec x
!
All these trigonometric derivatives
are based on the
angles measure being in radians.
3.
y = 3xex + x tan x
Find y', y'', y''', and y(4) for y = sin x
y' =
! ! !
4.
!
y''' =
!
y(4) =
2 x
f(x) = 3x e sin(x)
y'' =
!
If f (x) = sec x , find f ''(x).
tan x
,
x
(a) find the equation of the tangent to graph at x = 2.
!
!
!
(b) find the equation of the normal to the graph at x = 2.
sin 4x
=
x"0 sin 6x
lim
!
!
For f (x) =
sin 4x
=
x"0 3x 2 # 2x
lim