Page 94 - Foundations Of Differential Calculus

5

On the Differentiation of Algebraic
Functions of One Variable

152. Since the differential of the variable x is equal to dx, when x is
incremented, x becomes equal to x + dx. Hence, if y is some function of x,
I I
and if we substitute x + dx for x, we obtain y . The difference y − y gives
n
the differential of y. Now if we let y = x , then
I n n n−1 n (n − 1) n−2 2
y =(x + dx) = x + nx dx + x dx + ··· ,
1 · 2
and so

I n−1 n (n − 1) n−2 2
dy = y − y = nx dx + x dx + ··· .
1 · 2
In this expression the second term and all succeeding terms vanish in the
n
presence of the first term. Hence, nx n−1 dx is the differential of x ,or
n
d.x = nx n−1 dx.
It follows that if a is a number or constant quantity, then we also have
n
d.ax = nax n−1 dx. Therefore, the differential of any power of x is found
by multiplying that power by the exponent, dividing by x, and multiplying
the result by dx. This rule can easily be memorized.
n
153. Once we know the first differential of x , it is easy to find its sec-
ond differential, provided that we assume that the differential dx remains
n−1
constant. Since in the differential nx dx the factor ndx is constant,
n−1
the differential of the other factor x must be taken, which will be