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.

~ 129. General criterion. 223
F(x+ a) - F(x) Sz(x+ Ax)- z(x) R_, (x+ Ax) - R, (x)
1) A Ax Ax
For any finite value Ax however small, this continuous expression in
Ax has a determinate finite value; we can write it:
2) F(x+Ax)-F(x)x)
2) flx'( ++ A A-X f(2'x + Aoa X)+ f + - ( + + A X)
+B. (X + AX) - aB (x)
+ Ax
Now making A x converge to zero and denoting the differential
quotient of the remainder function n (x) by aB'(x) we get:
3) F'(x) = f1'(x) + f2' (x) +- f -l(x) + BR'(x).
If then the remainder of the original series be so constituted, that
for any number 6 however small, a place n can be assigned from
which onwards not only Rn (x) but also Rn'(x) remains smaller than 6,
however large n be, this equation passes over into the infinite series:
4) F'(x) f,'(x) + f2'(x) + -t- - f- (x) +- f,,(x) + - -.
The statement of the result is: If the remainder of an infinite
series possess the property, that for a given value of x by choice of a
lower limit for n, Irt'(x) becomes arbitrarily small, the series formed of
the differential quotients of the several terms is convergent and expresses
at this point the value of the derived function.
Since this property of the remainder in any arbitrary series cannot
be recognised at once, we may usefully establish another criterion, not
indeed necessary, but still sufficient, by which in many cases the
question is decided:
If the series of the derived terms converge uniformly in an interval,
it expresses everywhere in this interval the derived of the given series.
In order to examine this, let us denote the value of the supposed
uniformly convergent series
f )'(x) + f2'(X) + fn(X) +
by P (x) and its remainder from the uith place by Pn (x), then equation 2)
passes over into the form:
F(x + A x) - F(x) (x (x + AAx)-, \(x)
— Ax - ={(x+ eAx) —P(x+OAx)}+ Ax
Now here let n first become infinite and then put Ax = 0. When n
becomes infinite, the value of 0 also changes. But whatever be its
value, since the derived series converges uniformly, a value n can be
assigned, from which onwards we shall have P( (x + ) Ax) < 6 for all
values of 0. In like manner n can be chosen so great that the last
expression also shall amount to less than 6. Therefore a point can be
found in the interval from x to x + Ax, at which the continuous