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.

Sixth Chapter.
Differentiation of the simplest functions.
25. We shall first treat the functions defined in Chapter IV which
are styled Elementary functions.
I. The algebraic, whose simplest type is y == xn, and whose most
general is the implicit function: A0 ynl -A1 yi-il +... An_- y +-A, =0
in which A0, A1,... A, are polynomials of any degree in x.
II. The transcendental, namely:
a) the exponential function y = aa and the trigonometric: sin x,
cosx, tanx, cotx;
b) the logarithm y = alogx and the circular functions: sin-'x,
cos1 x, tan-1 x, cot-1x.
The immediate aim of our investigation is: from the properties of
these functions to obtain convenient methods of calculating them; for,
with the exception of the case y =xm for m a positive or negative
integer, in which the calculation is accomplished by carrying out
an n-fold multiplication, we have been hitherto put off with the process
of inclusion within limits or geometrical considerations. The most
comprehensive problem is presented by the implicit algebraic function,
but its treatment must be preceded by further general considerations.
26. For the formation of first differential quotients whether with
positive or negative values of Ax, the following General Rules are
required.
1) The differential quotient of a constant is equal to zero.
2) The differential quotient of a sum of functions is equal to the
sum of the differential quotients of the summands. For if
y = fi (x) f + 2 3( + ) +.. + fn (x),
then
y + Ay - fI (x + Ax) + f2 (X + Ax) +.. *+ fn (X + Ax),
therefore:
Ay fi(X+-Ax) —f (x) f2(x Ax)-f2(x) +.. _ f, (x+A) - n (X)
Ax Ax ar x Ax
Hence it follows by Proposition I ~ 10 that:
d x= -/ (x) +,, + * + f () Q. E. I).