# Differential Calculus for Beginners

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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1908 edition. Excerpt: ...is flog(-2)-log(a; + 2), and its nth differential coefficient We proceed to consider the various cases which may arise in the treatment of any proper fraction. If the denominator is of the fust degree, the fraction is already in its simplest form; e.g. 1 2 x+ 3: 3x + 4' If the denominator is of degree higher than the first, it can, in general, be broken up into factors. Several eases will arise, according to whether the denominator can be wholly broken up into real factors of the first degree or not, and whether any of its factors are repeated or not. For example, the denominators-1 )(x-2)(x + 3), (x-2)-(x + 3), x-+x+ 1, (x2 + x + 1 )3 are all of essentially distinct forms, and will require somewhat different treatment. 139. Unrepeated linear factors in the denominator. The first case which presents itself is that of a fraction whose denominator can be entirely broken up into real factors of the first degree, all different. We have to show how to find the simple fractions whose sum will be equal to the given fraction. Their denominators must, of course, be the several factors of the given denominator: the problem is how to find their numerators. The best way of showing the various methods of doing this will be by means of an example. US (x-l)(x-2)(x + 3) can be made up of the sum of A B, C x+x2+ x + 3' where A, B. C are constants whose values have to be such that when the three simple fractions are added together, the numerator shall be 2x--3. Now, if the fractions are added together, the resulting numerator is A (x-2) (x + 3) + B (x-1) (x + 3) + C (z-1) (x-2) = (A + B + C): e2 + (A + 2B-3C)x-(6A + 3B-2C). If this is to become 2a;2-3, we must have A + B + C = 2" A + 2B-3C = (U.-. A = i B=l, C =; 6A + 3B-2C = 3J 2g2-3 1/ 1 4 3 ..".