Page 99 - Foundations Of Differential Calculus

82 5. On the Differentiation of Algebraic Functions of One Variable
2 3
Likewise, if a + bx + cx + ex = y is a third-degree polynomial, then
2
dy = bdx +2cx dx +3ex dx
2 2 2
d y =2cdx +6ex dx ,
3 3
d y =6edx ,
4
d y =0.
In general, if the function is of degree n, then its differential of order n will
be constant, and higher-order differentials will all vanish.

161. Nor is there any difficulty with differentiation if among the powers of
x that make up a function we have negative or fractional exponents. Thus

I. If
√ c
y = a + b x − ,
x
then
bdx cdx
dy = √ + .
2 x x 2

II. If
a √
y = √ + b + c x − ex,
x
then
−adx cdx
dy = √ + √ − edx
2x x 2 x
and

2 3adx 2 cdx 2
d y = √ − √ .
4x 2 x 4x x
III. If
b c f
y = a + √ − √ + ,
3 2 3 2
x x x x
then
−2bdx 4cdx 2fdx
dy = √ + 2 3 3
√ −
3
3x x 2 3x x x
and
2 2 2
2 10bdx 28cdx 6fdx
√ +
d y = √ − 3 3 4 .
3
9x 2 x 2 9x x x