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.

Seventh Chapter.
Successive differentiation of explicit functions. Different orders
of infinitely small quantities.
32. The first derivate of a function or its first differential quotient,
as the calculations of last Chapter show, is itself again a function of
the variable. For the linear function y = ax + b, the progressive
and regressive differential quotient dy - a is constant, and this is the
only continuous function with a constant differential quotient, as we shall
prove in Chap. I of the Integral Calculus. Accordingly under similar
hypotheses further functions can be derived by the same rules from
each new derived function and their calculation always results from
what precedes. Let y =- f(x) denote the original function, further let
dy _ Lim f(x +Ax) f(x) f'(),
dx Ax
uniquely and determinately for each value of x, denote its first derivate,
then provided the function f'(x) is continuous and f(x+ Ax) — (x)
Ax
approximates to a determinate limiting value for Ax = 0, we obtain,
the second derivate: f"(x) = Lim f(x+ ) - () for Ax = 0
f A( x x)- f()
the third derivate: f"'(x) Lim f"(x + x)- f"(x) for Ax -0 etc..
Ax
These higher derivates as well as the first can be immediately
defined by means of the original function. For we have for Ax > 0
f(x~ + x) - f(x)
f'(x)= Lim f( + Ax)- f(
f(x + h7+ Ax) - f(x + h)
f(x + h) = Lim Ax
therefore:
"() -- Lim o Limjo f(x + h + x)- f(x + ) - f(x + Ax) + fx)
f"(x) = Lmj,,o Limd=O ------
f-(-) hAx
But this double limit can be determined more simply. We saw ~ 22
by the Theorem of the Mean Value, that for a continuous function
(p(x), the quotient of differences (x + Ax) - (x) can be always put
Ax
equal to the value of the derivate formed for a point x +- O Ax within
the interval from x to x + AX, where 0 denotes a number between 0