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3.5: The Derivative of Trigonometric Functions
Key facts:
Why is this true? For f (x) = sin x:
f (0 + h) − f (0)
h→0
h
sin h
= lim
= 1.
h→0 h
d
sin x = cos x
dx
d
cos x = − sin x
dx
f 0 (0) = lim
And for g (x) = cos x:
y
1
y = sin(x)
x
π
2
y = cos(x)
Or use the derivative plotter.
g (0 + h) − g (0)
h→0
h
cos h − 1 cos h + 1
= lim
·
h→0
h
cos h + 1
sin h − sin h
= lim
·
= 0.
h→0 h
cos h + 1
g 0 (0) = lim
We can use the angle addition
formulas for sin x and cos x to show
the general case (see pp 149, 150).
Derivatives of other trig functions:
Since tan x =
sin x
cos x
we have
At which x ∈ [−π, π] is there a
horizontal tangent in the graph
y = 2 sin x + x
d
d
sin x − sin x dx
cos x
cos x dx
d
tan x =
dx
cos2 x
Set the derivative equal to zero:
2
1
cos x + sin2 x
=
=
dy
cos2 x
cos2 x
= 2 cos x + 1 = 0,
if
2
dx
= sec x.
1
2π
cos x = − . Therefore, x = ± .
2
3
Similarly, since cot x = cos x/ sin x
sec x = 1/ cos x and csc = 1/ sin x,
y
d
cot x = − csc2 x,
dx
d
sin x
sec x =
= sec x tan x,
dx
cos2 x
d
cos x
csc x = − 2 = − csc x cot x.
dx
sin x
y=
2π
3
+
√
3
2
x
π
y = − 2π
−
3
√
3
Find the derivative for each of the following functions.
1
f (x) = √ cos x + 5x 2 tan x
x
1 d
d
1
d
d
0
√
cos x + √
f (x) =
cos x +
5x 2 tan x + 5x 2
tan x
dx
dx
dx
x
x dx
1 −3/2
1
=− x
cos x − √ sin x + 10x tan x + 5x 2 sec2 x.
2
x
For the following, use (uvw )0 = u 0 vw + uv 0 w + uvw 0 .
g (x) = xe −x cos x
d
d
d
g 0 (x) =
(x)e −x cos x + x (e −x ) cos x + xe −x (cos x)
dx
dx
dx
= 1 · e −x cos x + x(−e −x ) cos x + xe −x (− sin x)
= e −x (1 − x) cos x − x sin x .
Simple Harmonic Motion (pendulum or spring): s = A sin t.
Find the velocity and acceleration: s 0 = A cos t, s = −A sin t.