(a + =a" • + (mac + a.= + (male + + (malf + az,xtlnl + + m.0) +, &e. ; the law of which is evident, the only tbiog left being the substitution of the values in the tables instead of the derivatives of 6. This form is convenient for fractional or negative powers. The following case is worth exhibiting separately : 1 1 6 ac a+bx+,&e. - a - a3 x' Gs-nubs + a' xl + as xl We have avoided the formality of writing rib for c, IA for c, Cc.
A+B.c+cxl + , &C. s Ab - BCL = a - Agls -0c) -na b + ea' mob+ + ale)-na ae) + -Ea° - &e.
a4 The law is here evident enough ; the next numerator would be A(V- a - ant,' +ale) +cul(bl-ac)- + Fa*.
The derivatives of the general term 6' mny be readily formed, but the particular cases are more useful ; see the derivatives of a" in the general form above given. We shall not overload this subject with further examples : enough have been given to show those who require developments of some extent how much labour they might save.
We shall conclude this article by recommending that the process of derivation should be introduced, without demonstration, of course, into elementary books of algebra, as one of the best exercises of simple algebraical operation. We are firmly of opinion that the arithmetician and the analyst should be trained early in the performance of ope rations in which numerous details, each very simple in itself, follow one another in rapid succession with much sameness and some diversity. For this reason we should recommend, in arithmetic, Horner's process [INVOLUTION AND EVOLUTION] ; and in algebra, Arbogast's derivation. We proceed accordingly to divest this method of the phraseology of the differential calculus, and to put it before the elementary student in algebra.
The name of the process is derivation ; its primary object the raising of any power of an expression of the form 6+ cx + + f + , &e., immediately-that is to say, by writing down the result at once, without any but simple mental processes in passing from term to term. The rules are as follows : 1. Begin with that power of b which is to be raised.
2. To pass from the coefficient of one power of x to that of the next, multiply each letter by its exponent ; then diminish that exponent by a unit ; then introduce the next letter. And if this last process
increase an exponent, owing to the letter newly introduced having been in the term before, divide by the increased exponent. Bat remember nerer to operate on any letter except the last in the term, or the last but one ; upon the last always, upon the last but one when it immediately precedes the last in the original series 6, c, e, f, Ste.
3. If b+ ex + , &c., be not an infinite series, but a finite number of terms, operate as if the succeeding letters were severally equal to 0 : for instance, if g be the last letter, drop every term in which It should appear, as fast as it arises.
For example, the fifth power of b + cx + Begin with V, derive from it 5b•c, the two first terms arc . x.
To form the coefficient of take and observe that b and c follow each other in the series, so that in the next derivation there are two processes. First, use c or c', the last letter, which by the rule gives Ic"c or e : so that derivation applied to the first power of a letter gives merely a change of that letter into the next : hence gives But b's, which must also be used, gives Vilc, and gives 5(4bac)c; so that c becomes and we must therefore divide by the increased exponent 2, giving Hence the next term is (We In the next derivation gives only 51y, for b not immediately preceding c in the series b, c, c, &c., is not used. But gives 10(' c)cl 100(2ce)+ 3 , or 20bac.:.+ Next term (5bY In the next derivation 5/alf must be neglected entirely, because f is the last letter, and 6 is not the one immediately preceding. Also gives 20blef and or while gives 30b2c1c mad 2 s 106e. or The whole value of e.e t ext it as follows, and a little practice would enable any one to write it down at once, without any Intermediate operations :— This procese, so simple as compared with the actual performance of the four multiplications, has hitherto lain hid in works on the higher parts of the differential ealeidue : it is time it should take its place in every system of algebra which contains the binomial theorem, of which it is the legitimate extension.