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which result is the same as that obtained in Art. 29.

(16)

In the latter cases we have raised both sides to the square power, in order to avoid the apparent difficulty of taking the logarithms of negative quantities; but had this not been done, we should have had log, (-1), which will be shewn in the next Chapter to be equal to (2k+1) π √-1, the differential of which is zero, because it is a constant quantity.

Ex. 1.

Ex. 2.

y = (a+x) (b+2x) (c+3x),

dy

= dx (b+2x) (c+3x)+2 dx (c+3x) (a + x)
+3dx (a+x) (b+2x).

y = ex

... log. y =

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x + ¦ log (x + 1) − ¦
¦ log (x − 1),

= dx +

2

dy

y

.'.

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1 dx 1 dx
2x+1

dx

x2.

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dx

(x2-2) dx

2 x

.1

(x + 1) $ (x − 1) 3°

And a complicated product or quotient may often be most easily differentiated by first taking the logarithms, and then differentiating and reducing.

38.] Differentiation of (a) sina, (3) cos x, (y) tan x, (d) sec x, (e) versin æ, (8) cotx, (n) cosec x.

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dx

2

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dx

2

in accordance with Lemma II, Article 22, and omitting

which is added to x, we have

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Let the increments be infinitesimal, in which case tan de must be replaced by dr by reason of Lemma II, Art. 22; therefore

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omitting the term of the denominator involving dæ, because it is an infinitesimal subtracted from a finite quantity;

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= (sin x)-1,

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39.] It is also manifest from the geometry of the figure, see fig. 8, that the increments of the trigonometrical functions due to the increment of the arc are such as have been deduced in the above formulæ. For let AP be the arc of a circle whose radius is unity, and PQ be any small arc added to it, which ultimately becomes infinitesimal:

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ST: QP CT: CP, i. e. ST: Ax secx: 1.

.. ST Ax sec x.

When a becomes dx, PQ, ST become straight lines, and perpendicular to CT, and ST becomes da secx: whence

d. sin x = QR = PQ sin QPR PQ COS RPC PQ COS PCA

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d. secx ST' ST tan TTS ST tan PCA secx tan x dx.

40.] The differentials of tan x and cota may also be determined from those of sin x and cos x as follows:

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Ex. 5.

y = a cos x

(cosec x)2 dx.

y = (sin x + cos x)”,

dy = cos x2 2x dx.

dy = cos sin x cos x dx.

dy

=

-log, a.a cosa sin x dx.

dyn (sin x + cos x)"-1 (cos x - sin x) dx.

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