3.3 Derivatives of Trigonometric Functions
Keep in mind that angles will be used by the radian mode,
when you consider derivatives of trigonometric functions.
We can use the formula to obtain the derivative of sin x
sin x
= 1,
x →0 x
lim
cos x − 1
= 0.
x →0
x
lim
Indeed, they can be easily shown by the L’Hospital Rule.
Derivatives of Trigonometric Functions
d
d
sin x = cos x ,
csc x = − csc x cot x ,
dx
dx
d
d
cos x = − sin x ,
sec x = sec x tan x ,
dx
dx
d
d
tan x = sec2 x ,
cot x = − csc2 x .
dx
dx
Example1
1. Differentiate
1. f (x ) = x 2 sin x 2. y = 3 cos t + t 2 sin t
3. y =
x
2 − tan x
4. f (θ ) =
sin θ
1 + cos θ
5.y =
t sin t
1+t
2. Show that
d
csc x = − csc x cot x ,
dx
d
sec x = sec x tan x
dx
3. Find an equation of the tangent line to the curve at the given
point
y = cos x + sin x , (0, 1).
Example2
1. A mass on a spring vibrates horizontally on smooth level
surface. Its equation of a motion is x (t ) = 8 sin t.
(a) Find the velocity and acceleration at time t.
(b) Find the position, velocity, and acceleration of the mass at
time t = 2π/3. In what direction is it moving at that time?
2. Find the limit.
sin 5x
sin 3x
(b ) lim 2
x →0 3x
x →0 5x − 4x
sin θ
cos θ − 1
(c ) lim
(d ) lim
2θ 3
θ →0 θ + tan θ
θ →0
(a ) lim