Download (sin(x))

TRIGONOMETRIC FUNCTION DERIVATIVES
Sine Function Derivative
d(sin(x))
0
(sin(x)) =
= cos(x);
dx
• Also
d(sin(g(x)))
= cos(g(x))g 0(x).
dx
• Derivation: uses
sin(x + h) − sin(x)
h
sin(x) cos(h) + cos(x) sin(h) − sin(x)
=
h
sin(h)
cos(h) − 1
+ cos(x)
= sin(x)
h
h
with sin(small h) ≈ h, so limh→0 sin(h)
h = 1.
Also,
cos(h) − 1 (cos(h) − 1)(cos(h) + 1)
− sin2(h)
h
=
=
≈− ,
h
h(cos(h) + 1)
h(cos(h) + 1)
2
so limh→0 cos(h)−1
= 0.
h
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)?
c) g(t) = sin(t)/ sin(3t); g 0(t)?
r
2
TRIG DERIVATIVES CONT.
Cosine Function Derivative
d(cos(x))
= − sin(x);
dx
• Also
d(cos(g(x)))
= − sin(g(x))g 0(x).
dx
• Derivation uses
cos(x + h) − cos(x)
h
cos(x) cos(h) − sin(x) sin(h) − cos(x)
=
h
sin(h)
cos(h) − 1
− sin(x)
= cos(x)
h
h
• Cosine Examples:
a) y = − cos(8x2 + 2);
2
b) g(s) = cos(4e2s );
dy
dx ?
dg
ds ?
3
TRIG DERIVATIVES CONT.
Other Trigonometric Functions
• Use the quotient rule:
d(tan(x))
= 1 + (tan(x))2 = (sec(x))2;
dx
d(sec(x))
= sec(x) tan(x);
dx
d(csc(x))
= − csc(x) cot(x).
dx
• More Examples
dy
a) y = csc(x)/x; dx
?
4
TRIG DERIVATIVES CONT.
b) f (t) = sin(ln |2t3|);
df
dt ?
c) Runner armswing angle y(t) = π8 cos(3π(t − 31 ));
y 0(t), y 00(t), force?
5
TRIG DERIVATIVES CONT.
d) Alaska CO2 level in ppm, x = # of years past 1960.
C(x) = 330 + .06x + .04x2 + 7.5 sin(2πx)
Matlab
ezplot(’330+.06*x+.04*x*x+7.5*sin(2*pi*x)’,[0,50])
Alaska CO2 Levels
440
420
400
380
360
340
320
0
5
10
15
20
C 0(x)?
6
25
30
35
40
45
50