Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

OF SECOND, THIRD, &c. FLUXIONS.

..

13. The second fluxion of a quantity is the fluxion of the first fluxion. The third fluxion is the fluxion of the second, and so on: thus a signifies the fluxion of x; x, that of x. The square of the fluxion is written ; its mth power is written ", &c.

From what we have just said respecting first fluxions, it will be easy to find the second fluxions, &c. Thus if we require the second fluxions of r2; we shall have for the first fluxion 2xx; consequently the second fluxion will be 2xx+2xx=2x2+2xx. Similarly, since flux (x)=mxm-1x, we shall 2d-flux (xTM)=mxTM-1 x+m.m-1x=22 again flux (xy) = xỷ+yx; therefore 2d flux (xy)=xy+yx+2yx. Since flur x ___yx—xy__

xy, we easily infer that 2d flur

()

y yy

y yy

[merged small][merged small][merged small][ocr errors][merged small][merged small][ocr errors][merged small][merged small][merged small]

Qxya + y2x — xyÿ—2yxy, and in the same manner with other func

tions.

y3

By the same principles we are enabled to find the third, fourth, &c. fluxions, and in general the fluxions of any degree, of all sorts of quantities affected with x and y.

For example the fluxion of yr is y+y....that of √ (¿1+y')

[merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small]

14. In order to shorten the calculation of the second fluxions of several variable quantities, we generally suppose one of the first fluxions to be constant; that is, we refer the other fluxions to that one, as to a fixed term of comparison. We shall soon see examples of this process.

This kind of supposition simplifies the labour by causing all those terms to disappear which are affected with the fluxion of the quantity assumed as constant.

For example, if we required the fluxion of; on the supposition

that is constant, we should find the required fluxion to be ryzy

and if we considered y as constant, we should obtain

+

[ocr errors]

yx.

y

15. Hitherto we have supposed that the variable quantities of which the fluxions were required, increased all at the same time. If some increased, while others diminished, this would cause no difficulty, for x and y may be positive or negative, as well as any other algebraic quantities.

[ocr errors]

OF LOGARITHMIC AND EXPONENTIAL FLUXIONS.

16. Let it be required to find the fluxion of the natural logarithm of the variable x. We shall denote this logarithm by lx, and making lx=x, we have += (x+x); this gives z or flux (x)=l

(x + x) — lx = l(1 + =)==

the higher powers of x.

[blocks in formation]

Hence the fluxion of the logarithm of any quantity is equal to the fluxion of thal quantity divided by itself. Consequently in a system

mx But in

in which the modulus-m, we shall have flux (lx)= x

the subsequent articles we shall treat only of the natural or hyperbolic logarithms whose modulus=1.

By the above rule, we easily find flux (lx")

=

#c---- Nx &c.

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors]

17. If we had the powers of logarithms, or even logarithms of logarithms, it would be easy to find their fluxions. For example, let y=(lr)”, we shall have y=m (la)=-1= If we had y=x′′ (kz)",

we should obtain y=mxm1 x (lx)" + nxm-1 x (lx)"1 = xTM1 x (lx)~1 (n+mlx), &c. Again let y=llr; make r=z, and we shall have

[merged small][ocr errors][ocr errors][merged small]

==, ¿ =»

18. The equation flux ((x) === gives = flux (le). Consequently the fluxion of any quantity is equal to the product of that quantity by the the fluxion of its logarithm. This rule may be applied to facilitate finding the fluxions of quantities of any sort, even if algebraic. For example flux (x)=x flux lxTM=

mxml'x.

x y

Flux (zy)=xy( +)=y+y. Flux (

yx + xy
y1

mx x

[ocr errors]
[ocr errors]

The same principle may be applied with

success in finding the fluxions of exponential quantities; by which name is to be understood, such quantities as have variable exponents.

Such are a*, **, &c. which are of the first order; 2, which is of the second, &c.

The fluxion of a*, will be according to this rule a* flux la*=a* ffux (xla) = a*xla. Therefore if e be the number 2.7182818, of which the logarithm is 1, we shall have flux (e") = ex. Similarly

flux (x3) = xo flux (ylx) = x} (ylx + Ya), &c.

[ocr errors]

19. We might also have found these fluxions in the following We have seen that n = 1 + In + (ln)2 (ln)3 + + &c.

manner.

2

2.3

Suppose now that n=a*, and substitute this value in place of ʼn, and

2.3

we shall find a′′=1+la”+

(la), (la)3
+. + &c.

Now la*=xla, and

2

[ocr errors][merged small]

(la) = (xla)2 = xl'a; therefore a*—1+xla+

and consequently flux (a*) = xla + xx l'a +

[blocks in formation]
[ocr errors][merged small]

With respect to such exponentials as, their fluxion is easily found: for we have

flux (x'")=xo"flux (y' lx) = xo" {y =+ylx (zły +

zy

=y

+ZY lx + xlxly). If x=y=e, we shall have eez for the

I y

fluxion of e'.

In a similar manner we may find the second, third, &c. fluxions of logarithmic and exponential quantities, but it is needless to dwell longer on this head.

OF THE FLUXIONS OF SINES, COSINES, &c. AND OTHER CIRCULAR FUNCTIONS.

20. Let sin x=y, we shall have y+y=sin(x+x) = sin x cos z + sin x cos x. Now x representing an infinitely small arc, we shall have 1st, cos x=1; 2dly, sin x=x. Therefore y+y=sin + cos ■, or y= flux sin xx cos x. Consequently the fluxion of the sine of any arc is equal to the fluxion of that arc multiplied by its cosine. 21. Since flux sin x cos x, if we make x = 90°-y we shall have a-y, and flux cos y y sin y, a formula which we might have obtained in either of the two following manners.

[ocr errors]

Ist. Sin2 + cos2 = 1. Therefore sin x (flux sin x) + cos x (flux cos x)=0, and flux cos x = (flux sin x) =— - sin x

or 2dly, flux cos x =cos (x+x)

x sin x -cos x= r sin r.

-

sin r

[ocr errors]

cos = cos x cos x sin

Hence the fluxion of the cosine of any arc is equal to the negative fluxion of that arc multiplied by its

sine.

[merged small][ocr errors][merged small][ocr errors][merged small]

= flux tang x.

COS

2x

cos 2x + sin 2

cos 2x

Hence the fluxion of the tangent of any

arc is equal to the fluxion of that arc divided by the square of its cosine. If instead of supposing the radius = 1, we had supposed it = a,

[merged small][ocr errors][ocr errors]

23. Let x=90°-y, we shall have flux cot y =

[blocks in formation]
[merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small]

sin y

sin' y

sin y

y cot y. sin y

These same formula may also be found in the following manner.

Let the arc AB be denoted by z; its
cosine CD by ; its sine BD by y; and
the radius to be unity.
suppose
Conceive now AB to be lengthened
by the indefinitely small arc Bm, and
draw mr perpendicularly on BG; we
shall have Bm ¿, Br— — 2, and mr

=

E

[ocr errors]

DA

CD,

=y. The similar triangles CBD, Bmr, give
Bm: BC:: Br: BD :: mr :
or, * : 1 ::-: √(1—x3) :: y: √ (1—y1)

[merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

Similarly cos AB or = √(1-y'), and therefore or flux cos

[ocr errors]

y
√(1-y1)

——¿ sin z.

Let AT or tan z=t, we shall have t:1::y:√(1-y1), and con

[blocks in formation]

i

(1—y1)

Hence it follows that flux (4), or —— or flux cot 1 ——

[blocks in formation]

y'√(1—y3)

as before.

[ocr errors]
[ocr errors]

These rules enable us to find the first, second, &c. fluxions of any quantity in which there enter sines, cosines, &c.

Thus, for Example.

Flux (sin x)” = m (sin x)-1 x cos x-m sin x x cot x.

Second flux (sin x) = x cos x —

Second flux (cos x)

1

- sin x.

= flux (— x sin x) — — x sin a—¿a cos .

Flux (sin mx)= mx cos mx.
Flux (cos mx) =—mx sin mx.

« ΠροηγούμενηΣυνέχεια »