97. When quadratic factors (which are not resolvable into real linear factors) occur in the denominator, it is often convenient to make use of Demoivre's Theorem.

Ex. Let

y

(x+a)2+b2 {(x+a)+ib}{(x+a)—ıb}

then 9-(-1){ 2ib(~1)^n!

Yn

21b

(x+a-ib)2+1 (x+a+b)n+1

(−1)nn!f(x+a+ib)n+1 — (x+a—ib)n+1}

[(x+a)2+b2]n+1

n

× {(cos 0+ɩ sin О)"+1 — (cos — sin ()~+1}
(-1)n! 2 sin (n+1)0

Find the nth differential coefficients of y with respect to x in the

To find the nth differential coefficient of a product of two functions of x in terms of the differential coefficients of the separate functions.

d

cx

It was proved in Chap. II., Prop. IV., that

It appears from this formula that the operative symbol

or D may be considered as the sum of two operative

1

symbols D1 and D2, such that D, only operates on u and differential coefficients of u, while D, operates solely

upon v and differential coefficients of v. For with such

1

D1+D2:

Now, since D1 and D, are symbols which indicate differentiations, they each, like the original symbol D, obey the distributive and index laws and are commutative with regard to constants and each other. It therefore follows by formal analogy with the Binomial Theorem

It appears therefore from this formula that if all the differential coefficients of u and v be known up to the nth, inclusive, the nth differential coefficient of the product may at once be written down.

99. Another Proof.

From the importance of the above result it is considered useful to add here an inductive proof of the same theorem.

n

[Lemma. If Cr denote the number of combinations of n things r at a time, then will

nCr+nCr+1=n+1Cr+1•

This will form an easy exercise for the student.]

Let y=uv, and let suffixes denote differentiations with regard to x. Then y11+UV1,

Y1⁄2=UqV+QU1V1+uv2, by differentiation.

Assume generally that

= Un+1V+n+1C1UnV1+n+1C2Un-1V2+n+1C3Un - 2V3 +

+n+1Cr+1Un-rVr+1+...+uvn+1, by the Lemma ;

therefore if the law (a) hold for n differentiations it holds for n+1.

But it was proved to hold for two differentiations, and therefore it holds for three; therefore for four; and so on; and therefore it is generally true, i.e.,

(uv)n=Unv+nC1Un - 121+nC2Un−2V2+...+nCrUn-rVr+

100. Applications.

...

Ex. 1. Let y=exX, where X any function of a.

x.

Y1 = x3a”sin (ax+1T) +n3x2a”-1sin (ax+

n

2

2

(ax + 2 = 1 7 )

(ax+ n = 2 =)

2

3!

Since a number of examples on successive differentiation and on integration depend on the ability of the student to put certain fractional forms into partial fractions, we give the methods to be pursued in a short note.