Download TRIGONOMETRIC FUNCTION DERIVATIVES Sine Function

TRIGONOMETRIC FUNCTION DERIVATIVES
d(sin(x))
Sine Function Derivative : (sin(x))0 =
= cos(x).
dx
d(sin(g(x)))
= cos(g(x))g 0(x).
With chain rule
dx
• Derivation: uses
sin(x + h) − sin(x)
sin(x) cos(h) + cos(x) sin(h) − sin(x)
=
h
h
cos(h) − 1
sin(h)
=
sin(x)
+
cos(x)
h
h
sin(h)
with sin(small h) ≈ h, so lim
= 1.
h→0
h
cos(h) − 1 (cos(h) − 1)(cos(h) + 1)
− sin2(h)
h
Also,
=
=
≈− ,
h
h(cos(h) + 1)
h(cos(h) + 1)
2
cos(h) − 1
so lim
= 0.
h→0
h
2
TRIG DERIVATIVES CONT.
sin(x)
1
0.5
0
−0.5
−1
0
1
2
3
4
5
6
4
5
6
x
cos(x)
1
0.5
0
−0.5
−1
0
1
2
3
x
• Sine Examples:
a) f (x) = −3x sin(4x); f 0(x)?
b) f (x) = csc(x); f 0(x)?
3
TRIG DERIVATIVES CONT.
d(cos(x))
Cosine Function Derivative :
= (cos(x))0 = − sin(x).
dx
d(cos(g(x)))
With chain rule
= − sin(g(x))g 0(x).
dx
• Derivation uses
cos(x + h) − cos(x)
cos(x) cos(h) − sin(x) sin(h) − cos(x)
=
h
h
cos(h) − 1
sin(h)
=
cos(x)
− sin(x)
h
h
• Cosine Examples:
dy
a) y = cos(8x + 2); ?
dx
2
dg
b) g(s) = cos(4e ); ?
ds
2s2
4
TRIG DERIVATIVES CONT.
Other Trigonometric Functions
• Use the quotient rule with sin and/or cos rules:
d(tan(x))
0
(tan(x)) =
= 1 + (tan(x))2 = (sec(x))2;
dx
d(sec(x))
(sec(x)) =
= sec(x) tan(x);
dx
0
d(csc(x))
= − csc(x) cot(x).
dx
• More Examples
csc(x) dy
a) If y =
,
?
x
dx
5
TRIG DERIVATIVES CONT.
df
b) If f (t) = sin(ln |2t3|),
?
dt
c) Runner armswing angle y(t) = π8 cos(3π(t − 31 )).
Find y 0(t), y 00(t)?
d) Alaska CO2 level in ppm, x = # of years past 1960.
C(x) = 330 + .06x + .04x2 + 7.5 sin(2πx). Find C 0(x)?