ARTICLE XVIII.

Continued from page 138.

¶ 13. Hence it appears, that whether the root be a simple or a compound quantity, the fluxion of any power thereof is found by the following

RULE:

Multiply by the index, diminish the index by unity, and multiply by the fluxion of the root.

Examples.

Ex. 1. The fluxion of x is 9xx.

Ex. 2. The fluxion of 3y2 is 15y*y.

3 4 12

Ex. 3. The fluxion of — is

Ex. 6. What is the fluxion of a2 +x2 ]3 ?

35x

99x7T

Here the root is a2 + x2, and its fluxion 2xx; hence, the fluxion required is 3× a2+x2} 2 ×2xx= a2+x22 ×6xx.

Ex. 7. What is the fluxion of √a2+x2, or of a2+x2 = ?

Here the root is a2+x2, and its fluxion 2xx;

1

hence, the fluxion is-Xa2+x2) X2xx=

Ex. 8. What is the fluxion of x2+y2}} ?

Here the root is x2+2, and its fluxion 2xx+2yÿ;

3

I

hence, the fluxion required is×x2+123⁄4×2xx+2yj

22

➡3×x2+y2±2×xx+ÿÿ.

Ex. 9. What is the fluxion of x+}\2?

Here the root is x+y, and its fluxion x+y; hence, the fluxion required is 2xx+yxx+y•

Ex. 10. What is the fluxion of a3 +x31⁄2 ?

Here the root is a3+x3, and it's fluxion 5x+x;

hence, the fluxion required is

1

This quantity becomes a2+x2), and the root is +x2, whose fluxion is 2xx; hence, the fluxion re

like manner, bring any quantity from the denominator up to the numerator, by changing the sign of the index, and then proceed by the rule.

Ex. 12. What is the fluxion of ax2+by+cz4? Here the root is ax2+by+cz4, and its fluxion 2axx+3by2+4cz3%; hence, the fluxion required is

7

§×ax2+by3+cz11‡×2axx+3by2j+4cz3%.

Ex. 13. What is the fluxion of x2+√a2+y2? Put z=√x2+√a2+y2, then z2=x2+√ a2+y2 ¿

1

the fluxion of a2+y2, or of a2 +2, is ×

2

-\×?ÿÿ =a2+p2\—ž×ÿÿ; hence, 2zz=2xr

-2

14. The fluxion of x+y, by the last rule, is

2

2xx+yxx+y=2xx+2x+2yx+2yy; also,+y =x2+2xy+y2, whose fluxion is 2xx + the fluxion of 2xy+2yy; make these two values of the fluxion of x+2 equal to each other; omit the first and last terms which are common to both, and we have the fluxion of 2xy=2x+2yx; hence the fluxion of xy is xy+yx.

Otherwise thus. If we suppose a constant, the

fluxion of xy is xy by Prop. 3; and if we suppose y constant, the fluxion is yx; hence, if neither be constant, the fluxion is xy+yx.

For

Cor. Hence, we may find the fluxion of xyz. if v=xyz, and w=xy2 then vwz, and v=wz+zw; but w=xy, .w=xy+yx; substitute these values for w and w, and we get v=xyz+zxy+zyx.

15. In like manner we proceed for any number of factors; hence, the fluxion of the product of any number of quantities is found by the following

RULE:

Multiply the fluxion of each quantity into the product of all the rest, and the sum of all the products is the fluxion required.

Examples.

Ex. 1. The fluxion of x2y3 is x2×3y2j+y3 × Ÿxx =3x2 y3y+2y3 xx

7

Ex. 2. The fluxion of y‡æ3z is a3z×_j‡j+j3z×

x

Ex. 3. The fluxion of wxyz is mxnyrzswm-1w+ nwmyïz3xn−1x+rwmxnzsy2¬1y+swmx#yrzs−1%.

Z.

Ex. 4. To find the fluxion of x2xa+ya.

3

By the last rule, the fluxion of a++y^ is a++y4]ŝ ×4y3y=6×aa+y4\\xy3y; hence, the fluxion required

14

is x2×6×aa+y+\{×y3y+a++y4×2xx.

Ex. 5. To find the fluxion of ✔a+x2×√b2+y2. Find the fluxion of each part by the last rule, and

the fluxion required is √a2+x1×

yy

¶ 16. It appears from this Prop. that the fluxion of xy consists of two parts, xy and yx, the former part arising from the increase of y by y, and the latter from the increase of x by x; but if x should decrease whilsty increases, then the fluxion, expressing the increase of xy upon the whole, will be xy-yx, being the increase minus the decrease. Hence, to express the rate at which any quantity increases, the fluxion of the parts which increase must be written with the sign+, and those which decrease with the sign-. Now the increasing quantity is considered as positive; but if a negative quantity increase in magnitude, it must be considered as a decreasing quantity, and its fluxion will be negative. In like manner, a negative quantity decreasing in magnitude must be considered as an increasing quantity, and its fluxion will be positive. If therefore the fluxions of increasing quantities be writ

ten with the sign+, and of decreasing with-, when`ever the fluxion of any quantity is positive, it shows that quantity to be in an increasing state; and when negative, to be in a decreasing state.

x

17. Put z=-, then zy=x, and zÿ+yż=x (¶

y

From the fluxion of the numerator multiplied into the denominator, subtract the fluxion of the denominator multiplied into the numerator, and divide by the square of the denominator.