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.

~ 26. 27.

General Rules.

49

to u as independent variable, while that of the second is the differential
quotient of u with respect to x. Consequently we have:
dy df(t) dz
dx du dx
27. Explicit rational algebraic functions.
1) y = xm.
a) m a positive integer. - oo < x < + o.
For x = — co, the absolute value of the function also becomes
infinitely great; for every finite value of x, the function is continuous and
has a differential quotient finite in value; for, multiplying out we find
(X+ A x)m-xm _~ xm -X A x +2(A- )2+- C3 ( ~Ax)3 + CG (~ Ax)m
+Ax + Ax
The coefficients C are finite, depending on x and m, not on Ax; their
further determination does not here concern us. Accordingly:
Lim (x+~ xA =- m -xm + C2 Lim (+ Ax) + C3 Lim (+ AX)2.
+ AX
' - + C. Lim (~ Ax)m-,
therefore for Ax = — 0: dy _ (m) = nxm-, in particular: d(x) 1
dx dx -,in parilar dx
The value Lim(mxm1-) for x = oo is to be regarded as differential
quotient at the point x = co.
m) m a negative integer. - c < x < + o,
y X= Tx = ( =- - In > 0).
X~
For x 0, the absolute value of the function is infinite. By ~ 26 Rule 4):
d 1
dy _ (x) _ -1 _ _ _
d d x X2 - - x +1
The differential quotient likewise is co for x = O, while for x -- oo
both the function y and its differential quotient converge to zero.
2) y = ao xm + a1 xm-l + a2xm-2.. * + aCn-iX -+ am - A,
m integer > 0. For every finite value of x we have by Rules 2) and 3):
d y dA
d?=mo^8 ( - _ D71 + ( X )aIx; -'- '' a,2 + - a,-x d'
independent by Rule 1) of the additive constant am.
If y = (ax + b)m =_ Um, u =- ax + b, we calculate the differential
quotient by Rule 5) without having to expand the binomial:
dy dy due
dy dy du _ m um-1 a n ma( ax + b)m-1i
dx du dx c
3) The fractional rational function:
aoxm+ alxm- ~ +. + am- 1 + an __ A
Y B'
y boxn + baxn-l +.. + b_- u + -b~ B
HARNACK, Calculus. 4: