91, 92] 
DERIVATIVES AND INTEGRALS 
187 
directions at P, or two tangents. At any rate it has not one definite tangent 
or direction (see Fig. 43). 
Now let us suppose that the curve has (like the circle or 
ellipse) a tangent at every point of its length, or at any rate every 
portion of its length which corresponds to a certain range of 
variation of x. Further let us suppose this tangent never perpen 
dicular to the axis of x: in the case of a circle this would of 
course restrict us to considering an arc less than a semicircle. 
Then an equation such as (1) holds for all values of x which fall 
inside this range. To each such value of x corresponds a value of 
tan : tan \Jr is a function of x, which is defined for all values of 
x in the range of values under consideration, and which may be 
calculated or derived from the original function </> (x). We shall 
call this function the derivative or derived function of (f)(x), and 
we shall denote it by 
</>'(»• 
Another name for the derived function of </> (x) is the diffe 
rential coefficient of (¡> (x); and the operation of calculating 
cp'{x) from cf>(x) is generally known as differentiation: these 
names are by no means happily chosen, but the reader should be 
familiar with them. 
Before we proceed to consider the special case mentioned 
above, in which yjr = \tt, we shall illustrate our definition by some 
general remarks and particular illustrations. 
92. Some general remarks. (1) The existence of a derived 
function cf)'{x) for all values of x in the interval a^x^b implies 
that at every point of this interval </>(#) is continuous. For it is 
evident that 
lim [$ (x + h) — cf)(x)]/h 
cannot exist unless lim (f>(x -f h) = <£(#), and it is this which is the 
property denoted by continuity. 
(2) It is natural to ask whether the converse is true, i.e. 
whether every continuous curve has a definite tangent at every 
point, and every function a differential coefficient for every value 
of x for which it is continuous*. The answer is obviously No : it 
* We leave out of account the exceptional case (which we have still to examine) 
in which the curve is supposed to have a tangent perpendicular to OX: apart from 
this possibility the two forms of the question stated above are equivalent.