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.

48

Differentiation.

Bk. I. ch. VI.

S c holium: Two functions, which differ only by an additive constant, have the values of the differential quotient the same.
3) If y be the product of two functions y = p (x) g (x)
then Ay _ (p (x +- Ax) P (x +- Ax)- (p (x) * (X)
Ax Ax
The expression on the right can be replaced for every finite Ax by
yp (x + A-x) (x A - ) + AXx) -- (x) Ax
If (p and p be continuous and their differential quotients have determinate values at the point x, then, by Theorem II ~ 10 when Ax
converges to zero:
dy _ { cp(x).p (x) }
dr -- dg = = () x ' (x) + (x) ' (x).
This law can be extended to any definite number of factors.
Scholium: If a signify a constant, when y = af(x) we have
dy __ daf(x) __ x
dx dx a (x).
4) If y - ((x) then at a point where 4p (x) does not vanish,
)p (x)'
(x _ ( + Ax) (x) (X) P ( X + AxC) - q (x) P (x + Ax)
P ( +Ax) p () ( (x) p (x + Ax)
or:
Ax (VA () x +p (x) +Ax)- )( )) (x) (x Ax)
By Theorem III ~ 10 we have therefore:
dy _ () cp' (x) - (x). ' (x)
dx (qp (x))2
Scholium: If y =, then dY - -((x)
y (x)) dx -- (V (X))2
5) It is convenient to consider more composite functions in the
form: y = f (u), where u itself denotes a function of x.
For instance y (a x + b)m may be treated under the form:
y = um, where u = a x +- b; or y = sin (x2) as y= sin (u), where u =- 2.
In such a case when x increases by A x it first makes u change;
let the amount of the change be A u, then: Ay = f(u + Au) - f(u)
therefore =Aly _ f (u- Au)-f(u) _ f(u Au) - f(u) Au
Ax Ax Au Arx
If u be a continuous function of x, and f a continuous function of u,
then when Ax becomes zero, zAu also converges to zero (~ 19 d).
The limiting value of the first factor on the right is, as its form
shows, simply the first derived of the function f taken with respect