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.

~ 59. 60. The differential quotient at a multiple point.

105

way it may be defined, can be expanded in a series of powers. (Bk. II
Ch. IV. Bk. IV Ch. III).
60. We return to the question as to the differential quotients
at these singular points. We assert, the values of k: h calculated
from the above-mentioned quadratic or cubic equation present, when
dy
real, the different values of the differential quotient d- at such point.
This proposition is manifest, for;: h is a quotient of differences, and
its limiting value defines the differential quotient. It can however be
deduced yet otherwise from the original equation of definition:
dly _ a. af
dx ax ay
We consider first the following simple but important case,
(cf. ~ 19c): If in a quotient cp (x) numerator and denominator vanish
simultaneously for a determinate value x- a, cp and 4' may be any
continuous functions, not only algebraic, then (~ 10) this quotient
has a meaning only in so far as it is the limit towards which the
values at neighbouring points tend; thus
q? (C) = Lim m (a + h) or p (a) Lim cp (a- lh)
(a) 7,=o0 v (a + (h) (a) 9=o0 (a - h)
If now the Theorem of the Mean Value in its first form hold for
the functions 9p and ~, the following general rule is found for calculating the limiting value. Put
cp (a + )h) - ((a) = h (pa + ( h), or, as a9 (a) = 0, ( (a + $-h)= 7 q' (a + Oh),
iP (a -+-) - 4 (a)= h "' (a - 'r ), or, as 4 (a) -= 0, V (a + h) = h '(a +h h ),
then
p (a + 7z) _ '(a + Oh) d s ' (a + O h) ' (a)
and so Lim
t (aC + h) -v'C (Ca +, l h) 'n ho '' (a + - rh) - 'a (a)'
i. e. the value of the quotient P (x) at a point where numerator and
'4 (X)
denominator vanish simultaneously, is, on the hypothesis assigned,
equal to the quotient of the first derived functions of gp and 4 at this
point.
If cp' (a) and 4' (a) likewise vanish, but if the theorem of the
mean value is applicable in its extended form, we have
p (a + 1) - 2" (a + Oh), ip (a + h) = E" (a+ Eh),
therefore
p (a) Lim (a + h) _ Lir " (a + Oh) cp" (a)
Ti (a) h=o 0 (a + h) h=o "f (a + 'al) ' h ' (a)
This demonstration is not possible for the special case that a