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7. Differentiation of Trigonometric Function
RADIAN MEASURE. Let s denote the length of arc
AB intercepted by the central angle AOB on a circle of
radius r and let S denote the area of the sector AOB. (If s
0
is 1/360 of the circumference, AOB = 1 ; if s = r,
AOB = 1 radian). Suppose AOB is measured as a
degrees; then
(i) 2 and
csc u du cotu C
S r2
360
Suppose next that AOB is measured as radian; then
2
(ii) s = r and S = ½ r
A comparison of (i) and (ii) will make clear one of the advantages of radian measure.
TRIGONOMETRIC FUNCTIONS. Let be any real
number. Construct the angle whose measure is radians
with vertex at the origin of a rectangular coordinate
system and initial side along the positive x-axis. Take P(
x, y) on the terminal side of the angle a unit distance from
O; then sin = y and cos = x. The domain of definition
of both sin and cos is the set or real number; the range
of sin is –1 y 1 and the range of cos is –1 x 1. From
tan sin and sec 1
cos cos
it follows that the range of both tan and sec is set of real numbers while the domain
of definition (cos 0) is 2n 1 , (n = 1, 2, 3, …). It is left as an exercise for
2
the reader to consider the functions cot and csc .
In problem 1, we prove
lim sin 1
0
(Had the angle been measured in degrees, the limit would have been /180. For this
reason, radian measure is always used in the calculus)
This instruction material adopted of Calculus by Frank Ayres Jr 13
RULES OF DIFFERENTIATION. Let u be a differentiable function of x; then
14. d (sinu) cosudu 17. d (cotu) csc2 udu
dx dx dx dx
15. d (cosu) sinudu 18. d (secu) secutanudu
dx dx dx dx
16. d (tanu) sec2 udu 19. d (cscu) cscucotudu
dx dx dx dx
8. Differentiation of Inverse trigonometric functions
THE INVERSE TRIGONOMETRIC FUNCTIONS. If x siny, the inverse
function is written y arcsinx. The domain of definition of arc sin x is –1 x 1, the
range of sin y; the range of arc sin x is the set if real numbers, the domain of definition
of sin y. The domain if definition and the range of the remaining inverse trigonometric
functions may be established in a similar manner.
The inverse trigonometric functions are multi-valued. In order that there be
agreement on separating the graph into single-valued arcs, we define below one such
arc (called the principal branch) for each function. In the accompanying graphs, the
principal branch is indicated by a thickening of the line.
y = arc sin x y = arc cos x y = arc tan x
Fig. 13-1
Function Principal Branch
y = arc sin x 1 y 1
2 2
y = arc cos x 0 y
y = arc tan x 1 y 1
2 2
y = arc cot x 0 y
y = arc sec x y 1 , 0 y 1
2 2
y = arc csc x y 1 , 0 y 1
2 2
This instruction material adopted of Calculus by Frank Ayres Jr 14
RULES OF DIFFERENTIATION. Let u be a differentiable function of x, then
20. d (arc sin u) 1 du 23. d (arc cot u) 1 du
dx 1 u2 dx dx 1 u2 dx
21. d (arc cos u) 1 du 24. d (arc sec u) 1 du
dx 1 u2 dx dx u u2 1 dx
22. d (arc tanu) 1 du 25. d (arc csc u) 1 du
dx 1 u2 dx dx u u2 1 dx
9. DIFFERENTIATION OF EXPONENTIAL AND
LOGARITHMIC FUNCTIONS
1 h
THE NUMBER e = lim 1 lim(1 k)1/k
h h k 0
= 1 1 1 1 1 2.17828
2! 3! n!
NOTATION. If a > 0 and a 1, and if ay = x, then y = log ax
y loge x ln x y log10 x log x
The domain of definition is x > 0; the range is the set of real numbers.
y = ln x ax -ax
y = e y = e
Fig. 14-1
Rules of differentiation. If u is a differentiable function of x,
26. d (alogu) 1 du, a 0,a 1
dx ulna dx
27. d (lnu) 1 du
dx udx
28. d (au) au lna du , a 0
dx dx
29. d (eu) eu du
dx dx
This instruction material adopted of Calculus by Frank Ayres Jr 15
LOGARITHMIC DIFFERENTIATION. If a differentiable function y = f(x) is the
product of several factors, the process of differentiation may be simplified by taking the
natural logarithm of the function before differentiating or, what is the same thing, by
using the formula
30. d (y) ydu lny
dx dx
10. DIFFERENTIATION OF HYPERBOLIC FUNCTIONS
DEFINITIONS OF HYPERBOLIC FUNCTION. For u any real number, except
where noted:
eu e u 1 eu e u
sinhu 2 cothu tanhu eu e u , (u 0)
eu e u 1 2
coshu 2 sechu coshu eu e u
sinhu eu e u 1 2
tanhu coshu eu e u cschu sinhu eu e u , (u 0)
DIFFERENTIATION FORMULAS. If u is a differentiable function of x,
31. d (sinhu) coshudu 34. d (cothu) csch2 udu
dx dx dx dx
32. d (coshu) sinhudu 35. d (sechu) sechutanhudu
dx dx dx dx
33. d (tanhu) sech2 udu 36. d (cschu) cschucothudu
dx dx dx dx
DEFINITIONS OF INVERSE HYPERBOLIC FUNCTIONS.
sinh 1u ln(u 1 u2),all u coth 1u 1 ln u 1, (u2 1)
2 u 1
1 1 u2
cosh 1u ln(u u2 1), (u 1) sech 1u ln , (0 u 1)
u
1 u 1 1 u2
tanh 1u 1 ln , (u2 1) csch 1u ln , (u 0)
2 1 u u u
This instruction material adopted of Calculus by Frank Ayres Jr 16
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