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.

40

Differential quotient of a function.

Bk. I. ch. V.

The process of forming the differential quotient of y or its first
differential coefficient, or the first derived off with respect to x is briefly
called differentiation.
22. It is however not only at each point that the values of the
first derived of a function give us information respecting the manner
in which the function changes whether positively or negatively. We
are now going to demonstrate that the change of value of the function,
even in an interval of finite extent, can be measured by the values
of the first derivate. With this purpose in view we first prove the
L e m ma: When a uniquee function, whose progressive differential quotient
coincides with its regressive at each point within an interval from x = a
to x = b, has equal values at the extremities of this interval, there
must be in the interval at least one point at which the first derivate
vanishes.*) For, either, the function has everywhere the same value,
in which case it is constant and its differential quotient everywhere
zero; or, the function attains in at least one point within the interval
its greatest or its least value (~ 17). It may even undergo repeated
alternations of increase and decrease, one such it must have in any
case. If x1 be such a point, then in its immediate neighbourhood
f(x -- h) - f(x) will have the same sign as f(xi + 7-) - f(x).
Consequently the quotients
f(x1 - h) - f(x,) and f(x1 + h) - f(xi)
- h h
differ in sign, however small we choose the value of h. Now these
two quotients have the same limiting value, since by hypothesis the
progressive differential quotient and the regressive are identical; but
a positive and a negative series of numbers can have the same limit,
only when this limit is zero, therefore at this point f'(x1) = 0.
*) Such a function can be exhibited geometrically by tracing a curve of
the form fig. 3:
-1
Fig. 3.
it can also have points at which the tangent is parallel to the axis of ordinates.
The theorem asserts, what is manifest geometrically: that when the ordinates of
the extremities are equal, there is between them at least one point with the
tangent parallel to the axis of abscisss.