Page 111 - Foundations Of Differential Calculus

94 5. On the Differentiation of Algebraic Functions of One Variable
If several factors occur, this kind of special rule can easily be worked out,
so it is superfluous to say more.
169. The rules for differentiating that we have presented so far are suffi-
cient to cover any algebraic function of x. If the function is a sum of powers
of x, this has been treated in paragraph 159; if the function is a quotient
of such functions, we have shown how to differentiate in paragraph 165.
We have also given an outline of differentiation when the function involves
factors. We have also taught how to differentiate irrational quantities, how-
soever they may affect the function, whether through addition, subtraction,
multiplication, or division. We are always able to reduce the function to
cases already treated. We should understand that the reference is to ex-
plicit functions. As to implicit functions given by an equation, these we will
treat later, after we have taught how to differentiate functions of two or
more variables.
170. If we carefully consider all of the rules we have given so far, and
we compare them with each other, we can reduce them to one universal
principle, which we will be able to prove rigorously in paragraph 214. In
the meantime it is not so difficult to see intuitively that this is true. Any
algebraic function is composed of parts that are related to each other by
addition, subtraction, multiplication, or division, and these parts are either
rational or irrational. We call those quantities that make up any function
its parts.
We differentiate any part of a given function by itself, as if it were the
only variable and the other parts were constants. Once we have the individ-
ual differentials of the parts making up the function, we put it all together
in a single sum, and thus we obtain the differential of the given function.
By means of these rules almost all functions can be differentiated, not
even excepting transcendental functions, as we shall show later.

171. In order to illustrate this rule, we suppose that the function consists
of two parts, connected by either addition or subtraction, so that

y = p ± q.

We suppose that the first part p is the variable part and that the second
part q is the constant part, so that the differential is equal to dp. Then we
suppose that the second part ±q is the only variable, while the other part p
is constant, so that the differential is equal to ±dq. The desired differential
is put together from those two differentials, so that

dy = dp ± dq,

just as we have seen before. From this it must be perfectly clear that if the
function consists of several parts conjoined by either addition or subtrac-