92] 
DERIVATIVES AND INTEGRALS 
189 
place in time, so that M moves along the axis of x: it is most 
convenient to suppose that it moves steadily in the same direction 
with uniform velocity V, so that at time t, OM = x — Vt. 
Let us suppose that t + r is 
a time a little after or a little 
before t, according as r is positive 
or negative: and let M', N', P' 
denote the corresponding posi 
tions of M, N, P. Finally let 
OM' = x + h, so that h = Vt, and 
let JVP cut M'P' in R. 
Then if (f){x) is a function 
such as those which we have been considering, the ratio RP'/PR 
converges to a limit, say l, when l is in fact the same as 
tan ^ in | 91. 
Also NN'jr = RP'/t = (BP*/PR)(PR/r), which has the limit 
VI when r-^-0. But the limit of NN'¡r is precisely what is meant 
by the velocity of N. Thus the value of the derivative (f>'{x) 
represents the velocity of N compared with the velocity of M. If, as 
is simplest, we suppose that M is moving with unit velocity, then 
<j>'{x) is actually equal to the velocity of N. Another way of 
expressing the same thing is to say that <£' (x) represents the 
rate of increase of (f (x), taking the rate of increase of x as our 
standard ; if <£ (x) is decreasing, this ‘ rate of increase ’ is of course 
negative. 
It is evident that the geometrical apparatus, by means of 
which we have arrived at this notion of the meaning of <f>'{x), is 
not essential. There is no reason why we should imagine the 
values of y represented along a line perpendicular to OX: this 
representation was adopted merely to show that our present point 
of view is not essentially distinct from that of § 91. Another way 
of looking at the matter is as follows. Taking V — 1 as above 
we have x = t: and there is no need for any special geometrical 
representation of x at all: x is the time t, simply, and we may 
regard the time t itself as the independent variable. We may 
simply take a line L and suppose the values of y — (f{t) represented 
by the lengths OQ measured along this line from a fixed point 0. 
If, at time t, OQ = <p{t), </>'(t) is the velocity of Q at time t.