Lagrange calls the function f x the primitive function, in respect of the functions fe x, f" x, Ste. These, again, in respect of the primitive function, he calls derivative func tions (functions derivees.) The function f' x is called the first derivative function, or derivative function of the first order, or simply the prime function ; the function f" x, de rived from it, is called the second derivative function, or de rivative function of the second order, or simply the second function ; and again, x, derived from the preceding, is the third derivative function, or derivative function of the third order, or third function, and so on.
Any function whatever, in respect to that from which it is derived, is its derivative function, and this last is the pri mitive function of the other.
Sometimes, instead of using the characteristic letter f, a function of x may be denoted by a single letter y; then, y being used instead of the symbol f x, the symbols y', y", y"', Sce. may represent the characters f' x, f" x. x, Ste. Ac cording to this notation, y being any function of x, when .r becomes x+h, then y will become y+i ' 2 ' 2 . 3Since every derivative function of the first order is mere ly the co-efficient of i in the developement of the primitive function f x, when x+i is substituted instead of x, the de termination of the derivative function of any power what ever x" is in fact the same thing as the determination of the term that contains the first power of i in the clevelopement of (x--1-i)" , according to the powers of i. Now it may be demonstrated by the elementary operations of algebra, that whether n be positive or negative, whole or fractional, the two first terms of the developement of (x-Fi)" are x" n i, (See ALGEBRA, art. 319 ; also FLCTXIONS, art. 7.) ; therefore, the first derivative function of x" is n It is now easy to find all the terms of the development of f (x-f-i)=(x +On . For since from f x=x" , we have f' x =n from this last we derive f" x =n (n-1) and hence again f"' x=n (7i-1) SIC. So that from the series f (x+i)= f x+if' xf" Ste.
2 we get n-1) (x +On =xn -F- Ste.
which is Newton's binomial theorem.
Next let the function be f x=ax , a being supposed con stant, and x variable ; then f Now, the com mon principles of analysis are sufficient to prove that the two first terms of the developement of axfi are as +A axi; here A is the Napierian log. of a, (see ALGEBRA, art. 355 ; also FLumoNs, art. 14. and 19.) Therefore the first deri vative'function of ax is A ex, that is, j'x = A (i• ; hence &c. These values substi tuted in the developement off (x +0 give A' .A" aa = + A al i+ — + .
2 .3 Let the function be f x = log. x, then f +i) = log. but it may be proved, as in the former cases, that the two first terms of log. (x+i) are log. be inv, put for the Napierian log. of the basis of the system ; (see FixxioNr, att. 18, and 19.) Therefore the first devi vative function of log. x is .----; and, because f' a' = 1; by the rule for the derivative function of a power, we hence foaltB x2 ; and again fm - These substitutions being made in the general B d•velopemcntcof f (x +0, we gut log. (x+i) = log. x + 3 ' B x ' 2 B B &c.
It has been shewn, (FLuxioxs, art. 17, and 19,) that the two first terms of the developements of the sine and cosine of x+i arc sin. (x +i) = sin. x-Fi cos. 2.4- &c.
cos. (x+i) = cos. x—i sin. x+ &c.
Hence it appears that the first derivative function of sin. x is cos. a:, and that the first derivative function of cos. x is — sin. x : Since therefore in the case off x= sin x, we have f' x = cos. x, it follows that f" x=— sin. s, f" — cos. x, &c. and since when F .r = cos. x, we have F' — sin. x, it follows that F" x= — cos. x, F" x= sin. x, &c. These expressions substituted in the development off (x+i) and F (x+i) give sin. (x+i)= sin. x+i cos. x— cos. 3-+ &c.
2.3 cos. (x-i- 0 = cos. x—i sin. x— cos. X +— X + 2 2 From the brief view we have given of this calculus, its intimate analogy with the method of fluxions, or differen tial calculus, must be evident. In fact, they all rest upon the same analytical principles, and the object presented to the mind in each is the snme ; for the different orders of de rivative functions in Lagrange's calculus are identical with the successive differentials, or rather differential co-effici cuts, in that of Leibnitz, and with the different orders of Iluxions in Newton's theory. The peculiarity of each cal culus, as delivered originally by the inventor, consists in that relation between the original function and its prime function, or differential, or fluxion, which the mind selects 'as a subject of contemplation. We have seen that it is a Fundamental proposition in analysis, that if x-f-i be substi tuted for x in any function fx, its new value f (x+i) has al ways the form f x+i r+ q, r, ac. being functions of x, winch are independent of i. Newton ob :,erved, that if x andf x are represented by two lines gene rated by motion, and if i be the velociiv of the point which generates s, then i/i, the second term of the developement, will he the velocity of the point that generatesf x ; (FLux roxs, art. 20-22.) hence he called in the fluxion of the 'unction f a-. Leibnitz again considered, that if x was in creased by the quantity i, then fx was augmented by the increment r+ &c. But supposing i indefinitely small, the first term of this series is indefinitely greater than the sum of all the following terms ; therefore, rejecting these, and retaining the term ip alone, he called it the dif ferential of the function fx. (1.1LxioNs, alt. 107-11u.) Lagrange, regarding the generation of algebt sic quantities by motion as incompatible with the principles of pure ana lysis, and also considering the doctrine of infinitely small quantities as too slippery a foundation For so sublime an edifice, he rejected both views of the subject, and deduced its principles from the theory of the developement of func tions into series.
It is in general admitted, that the Theory of Analyttc Functions has fulfilled the promise of its illustrious author, " to deliver the principles of the differential calculus disen gaged from the consideration of infinitely small or vanish ing quantities, also limits and fluxions." \Ve think, how ever, that he has under-rated the value of the theory of limits, as delivered by 'Maclaurin and D'Alembert, when he says that the kind of maaphysique that must be employed in it is, if not contrary, at least foreign to the spirit of analysis, which ought not to have any other metanhysiyue than that which consists in the first principles, and the first funda mental operations of algebra.
The ingenious author, in the discussion of his theory, has adopted a new notation. This has been matter of re gret, (Lacroix Cal. Di!: vol. i. at t. 82, 83.) because the no tation of the differential calculus was quite sufficient. In the comparison of methods and formulze, different notations are perplexing, and the number of arbitrary characters al ready employed in analysis, is a considerable and increasing evil. This, however, is but a small defect, when the lumi nous views and original methods which the work contains are taken into account. Many of the French mathemati cians regard the publication of the Theory of Functions as an era in analysis: Indeed, all the works of the differential calculus that have since appeared, have more or less adopt ed its views. See, in particular, Gamier, Lecons de Cal. Diffcrentiel. The mathematical reader will of course study the Thecrie des Fonctions itself; and it may be useful to know, that the author published a second and improved. edition of the work in 1813, a short time before his death. (g)