By Gert K. Pedersen

ISBN-10: 0387967885

ISBN-13: 9780387967882

ISBN-10: 3540967885

ISBN-13: 9783540967880

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For dxK ' dxK dx dx ' d(xn) dx (p. 26) •= (n+l)xn. If the result is true for any particular value of n, then it is true for all subsequent values. But it is true for n = 0, since Hence it is true for n = 1; hence for n = 2; hence for n = 3; and so on. (ii) Suppose that n is a negative integer. Write n — — m, so that m is a positive integer. x ') = dx ' =dxi-K (1) = 0. [The reader may wish to remind himself from a text-book on algebra about the rules for indices. The basic rule is that THE DIFFERENTIAL COEFFICIENT Hence 31 a r ^ + ^ ^ p = 0, ax or ax d(x\ x-™.

8x^0OU$x^0OX (Ju (assuming that -=- exists, as is implicit in dx the enunciation). Q ox dydu ~~ du dx' COROLLARY. The differential coefficient of the quotient u/v is vu' — uv' V2 y = ujv = iuv~\ Let du dx dy dx Then, as above, y—1 -\-11 • du 1 V' "~ dx TLT TTTi 1 . Note. When we write V*' u dv v2dx Idu vdx dy dx so that d(v~i )dv dv dx' -f U • d(% dv But (p. 21) ) dx 8y vuf — uv1 v2 8y 8u -^ = -^---, 8x 8u8x we have in mind that the increment 8u is not zero. But we know that u is the given function f(x), and so there will usually exist COEFFICIENT OF 'FUNCTION OF A FUNCTION* 29 (isolated) values of x at which/'(x) = 0; that is to say, the limit of Su -s- Sx is zero at such points, and so the initial step of dividing 8y by Su is open to the suspicion of being division by a zero denominator.

29. coseca:. 33. cos 2a;°. ty(sinx). DIFFERENTIAL COEFFICIENTS OF HIGHER ORDER 35 9. Differential coefficients of higher order. If f(x) is a given function of x, its differential coefficient f'(x) is another function of x, having in general its own differential coefficient. This is called the second differential coefficient of f(x)> and is denoted by the symbol »,,. x Alternatively, if y = f(x), the second differential coefficient of y is written in one or other of the forms „ dtf' y • In the same way, the differential coefficient of/"(#) is the third differential coefficient of f(x); and so on.

### Analysis Now by Gert K. Pedersen

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