Page 98 - Foundations Of Differential Calculus

5. On the Differentiation of Algebraic Functions of One Variable 81
so that its differential is equal to

dp + dq + dr + ds + ··· .

Hence, if we can give the differential of each quantity p, q, r, s, then we
know the differential of the sum. Furthermore, since the differential of a
multiple of p is the same multiple of dp,wehave d.ap = adp, and the
differential of ap + bq + cr is equal to adp + bdq + cdr. Finally, since the
differentials of constants are zero, the differential of ap+bq+cr+f is equal
to adp + bdq + cdr.

159. In polynomial functions, since each term is either a constant or power
of x, differentiation according to the given rule is easily carried out. Thus
we have

d (a + x)= dx,
d (a + bx)= b dx,
2
d a + x =2x dx,
2 2
d a − x = −2x dx,
2
d a + bx + cx = bdx +2cx dx,
2 3 2
d a + bx + cx + ex = bdx +2cx dx +3ex dx,
2 3 4 2 3
d a + bx + cx + ex + fx = bdx +2cx dx +3ex dx +4fx dx.

If the exponents are indefinite, then

n n−1
d (1 − x )= −nx dx,
m
d (1 + x )= mx m−1 dx,
n
m
d (a + bx + cx )= mbx m−1 dx + ncx n−1 dx.

160. Since the degree of a polynomial is given by the term with the highest
power of x, it is clear that if differentials of such functions are continually
taken, the differential will eventually become constant and then vanish,
provided that we assume that dx is constant. Thus the first differential of a
first degree polynomial a + bx, bdx, is constant, and the second and higher
2
differentials vanish. Let a + bx + cx = y be a second-degree polynomial.
Then

2 2 3
dy = bdx +2cx dx, d y =2cdx , d y =0.