And, since this is true, when (m) is a whole number, it will

also be true when (m)

n

is a fraction; therefore it is generally

The same form will hold true, if the index be negative.

Let (k) be the index, and let (m) and (n) be two numbers,

where A, B, C, &c. are the coefficients, whose value would have been found by actual division. Now the form of these coefficients is independent of the values of (m) and (n), and will, therefore, remain the same, whether (m) be greater than (n), or (n) than (m). Now, if (m) had been greater than (n), (m-n) would have been positive, and, consequently, we should have had

The

same form will, therefore, be true, when (n) is greater than (m); hence we have, [writing -k for (m-n)],

which form will hold good, whether (k) is a whole number or a fraction.

Ex. 9. d.xy3 = 2y3 xdx+3x2ydy.

CASE 2. Suppose the root to be a binomial (a+bx"), one of whose terms (a) is a constant quantity, and let (n) be the power to which it is raised, then

d (a + bx")"=n. (a + bx")"1× rb x-1dx.

Let u = (a + bx")", and y=a+bx” ;

Here then, there are three quantities, u, y, and x, du and the differential coefficient of (u) considered dy'

as a function of (y), is equal to n.(a+bx")"-', and

dy

dx

the differential coefficient of (y) considered as a

1

du function of (x), is rbx*-', and, therefore, the dx

differential coefficient of (u) considered as a function of (x) is, (by Prop. 21.)

and, therefore,

= n(a + b)-xrb -1,

I

du=n. (a+bx2)”-1× rbxˆ ̄1dx.

Ex. 1. d. (a + x)5

= 5.(a+x)1. dx.

= √ b2 − x2. × d.√ a2+x2 + √ a2 + x2. d √ b2 — x2

=

= (a3 — x2) ̄ ̄ 3. d . (a2 + x2)§ + (a2 + x2)1× d. (a2 — x2) — §

CASE 3. Next the root be a multinomial of the form

a+bx+cx2+ &c.

where (r), (s), &c. are either positive, or negative, integer or fractional numbers, and let (n) be the power to which this multinomial is raised, then