4.12 The differential 119
Derivatives with respect to time are sometimes written in a ‘dot notation’:
6(4.27)
0 Exercise 87
Angular motion
Consider a body moving in a circle of radius r. Let O be a
fixed point on the circle and let the position of the body
at time tbe given by the angle θ(Figure 4.13). The rate of
change of θwith respect to time is called the angular
velocity:
(4.28)
In addition, because arc length sis related to subtended
angle θbys 1 = 1 rθ, we haveK1= 1 r 4 or
v 1 = 1 rω (4.29)
for the relation between linear and angular velocities for motion in a circle (see
Chapter 16 for a more general discussion of velocity in terms of vectors). The angular
acceleration is 71 = 15.
0 Exercise 88
4.12 The differential
The first derivative of a functiony 1 = 1 f(x)is defined by equation (4.8),
and, as has been emphasized before, the symboldy 2 dxdoes not mean the quantity dy
divided by dx, but represents the value of the limit; in this sensef′(x), or y′, is a better
symbol for the derivative. It is nevertheless tempting to write
dy 1 = 1 f′(x) 1 dx
and, when properly interpreted, this is a useful way of describing changes.
dy
dx
fx
y
x
fx x
xx
= ′ =
=
→→
( ) lim lim
()
∆∆
∆
∆
∆
00
−−
fx
x
()
∆
angular velocity==
∆
∆
==
→
ω
θθ
lim θ
∆t t
d
0 dt
vv== , ==
xdx
dt
x
dx
dt
226In his 1671 paper on the calculus, Methodus fluxionum et serierum infinitorum, Newton considered variables
like xand yas flowing quantities, or fluents, and wrote Hand Ifor their rates of change, or fluxions. Dotted fluxions
were still being used by English mathematicians when the Cambridge undergraduates George Peacock (1791–1858),
John Herschel (1792–1871), and Charles Babbage (1792–1871) founded the Analytical Society in 1813. One of the
aims of the Society was to promote ‘the principles of pure d-ism as opposed to the dot-age of the university’.
..........................................................................................................................................................................................................................t
t+∆ t
θ
∆ θ
o
r
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..............................................................................Figure 4.13