An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

36

Differential quotient of a function.

Bk. I. ch. V.

I.
II.

Ay is the difference of altitude of two points P and P1, belonging
to the values x and x + Ax.
The quotient of differences is equal to the trigonometric tangent
of the angle, made by the chord PP1 with the axis of abscisse;
it measures the mean intensity of rising.
The equation of the chord, i. e.
the right line passing through
2Ij Hi^ the points P and P-, if g and
IKl a y X signify the coordinates of any
-/ I I point on it, is

It
I
Act AW..,I,
/6\~~~~ /A_ _JI
i
IB
N, I 11

- f(x) _ f(x Ax)-f(x)
e -x Ax
The point on this line with the
abscissa =- x + OAx has the
ordinate

'/ /\I? ~9A AA.

AA

/ / V].. fV1 I= Xf(x) +O ( f(x + Ax)-f(x).
Fig. 2.
F~ig. 2. On the other hand, to the abscissa x + 0 Ax belongs the vertex of the polygon

'q == f(x+0A O X).

Accordingly

{ f(x + Ax) - f(x) - {f(x + OAx) - f(X)}
is equal to the difference MTT'- MTT = TTrY'. Let us call TMf- the
measure of the deviation of the value of the function from the right
line at this point in the interval Ax, then the above inequality, which
includes the first mentioned condition, teaches us:
The necessary and sufficient condition that the quotient of differences may have a determinate finite limiting value, consists in it
being possible to find an interval Ax, in which the deviations from
the right line are in absolute amount less than any number 6 however
small. *)
Denoting by a' the angle which the chord PP1 makes with the
axis of abscissae and likewise by a" that which PTT makes, we obtain
another interpretation of the inequality: for
tanc '- tan a" f(X+ ) - f) f x + O-) f f(x) <
Ax OAx
that is, it must be possible to find an interval Ax, in which the
differences of the angles a', a" become arbitrarily small. The sides
of the polygon then approach to a fixed limiting position and the
*) Here the deviations ican continually change sign, i. e. a. curve can in any
interval however small have infinitely many oscillations relatively to its tangent.