130 



♦ KNOWLEDGE 



[July 21, 1882. 



^ir iflatOnnatiral Column, 



EASY LKSSONS IN THE DIFFERENTIAL CAIXIIAS. 

 No. IV. 



WE ore now to consider tlio differential coefficients of certain 

 familiar expressions, and to lay down rules by which cho 

 T r. : ;m1 !■ > ilii iont of any expression can be readily dctcniiined. 

 . Ut nie note that, for convenience, the nuantity 

 I jotliiicnt is to bo determined is commonly 

 .■ and the quantity whose variation causes tho 

 .- commonly expressed by r. 1 select ;; for the 

 i.irn.. r :itnl ivi.uii a fur the latter purpose. Let it be remembered 

 that then- is no real necessity for any lixed practice in this matter. 

 In tho last lesson, for example, I used in ono case s and ( instead of 

 ;; and i, while in the other example I interchanged x and ;/. 



I.pt me rei)eat that tho differential cocflicieut of ono quantity ;/ 

 with respect to another x, is an expression indicating tho rate at 

 which the former incTtaees as we increase tho latter. 1 advisedly 

 \\<» the wonl inf rcn-ie as respects \i, even though increase of .i' may 

 canso decrease of \i. For in snch a case tho differential coollicicut 

 will come out ncjolire, and a negative increase is tho true alge- 

 braical equivalent of decrease. 



And now for the differential coefficients of tho simple function. 

 If \i = a, oris constant, its differential cooDicient vfith respect to 

 IS of conrse 0. 



I>>t y=«", where » is a positive whole number. In this case we 

 will po through the process for finding tho differential coefficient. 

 We increase x by A i and assume that i/ is thus increased by A i/.* 

 Then ' - 



y + A !, = (^ + A x)- - • 



f a finite number of terms iuvolv- 



* I, Ir.L' fA .-V. fA ^■)^ Ac. 

 and 1/ = X* 

 Then-fore subtracting 

 Av — n i"~' Ai + torma involvin;; 1 i^ . < Ji . thnn the 



first. 

 Hence. 



_y- = ni"~' + terms involving Aj and. its powers. 

 A« 



Now suppose Ax, and therefore Ay, to become indefinitely 

 minute, calling them res|)Cctivcly d x and d y, and wo have 



<Ii/ _ .-1 . < * finite number of indolinitoly 

 _ —nx -^ miouto qoantities. 



The pxtonsion of this proof to tho cases where n is frnctimuil or 

 negative, or both, would not suit these colomns. Let it gulHce that 

 for all values of n 



if v-i- 



dx 

 Bnt let ns remove this result from the rejn'on of algebraical 

 'iprcssions for a moment, in order that its significance, and the 

 significancf of sequent results, may bo fully recognised. Take ' to 

 be 10, and ri to be 4, then 



y = 10' -1000. 

 Trv now the effect of adding to y somo small qnantity, nny 

 l-lOOth, or-01. Then instead of 10', y becomes (1001)*, and if 

 wu calculate tho value of this, we find it to bo 

 1004006001001 

 Tho effective part of the increment of y is tho -10, and fho pro- 

 : 'rtion it boars to the increment of x (or 1-lOOth) is 1000. This 

 ' » limes the cube of 10, or ni"-*. .\nd so in Any cafO the 

 •••uicr may care to try : whenever y — i", a minute increase in the 

 . alao of s gives to y an increase n i'~' times as great. 



The general result enables ns to at once express tho differential 

 rr,efficcnt of all such quantities as \/2>'^~ '''"' *° ""■ ^^ '^ 



only necessary to write these (or conceive them written), in tho 

 forms xi.r4.x^,t«>see that the respoctivo differential coefficients arc 



>')rtant to remember that A X is a single quantity, 



. X, bnt tho increment of x. In thf old nnta- 



-implo expression was nned. Hut in a<lvaiicod 



■ differential calculus, the notation of lliixions 



f:ii|.' t'.-.illv, ,•,:,>! ihc elx's and e/y's olone sirve tho mathematician's 



, and • 



I'll we may write 



Any ono who has become at nil practised in 



3 v., 



applying tho differential calcnlns, would of conrso write down 



Uieso results at once. It is plain, too, from the nio<l<> of proof that 



if y = oj», where a is constant, -r- •■ an .t"~'. 



* Tho next simple function I shall take is tho sine of an angle. 

 Anil having in \-iew tho importanco of tho reader's obtaining clear 

 views of Iho nature of a differential coefficient, 1 shall in this case 

 employ a geometrical way of finding such a coefficient. 



Lot tho angle .x be repiescntod by AOU, Kig. 1 ; then using the 

 arc measure and making tho radius unity, wo have 



» = arcAB, 

 and sin. x = BF, where BF is perpendicular to OA, 



=V suppose. 

 Now let angle AUB receive tho small increment BOC, and call the 

 arcBC, A.ti; then, completing the figure, the corresponding incre- 

 ment of the sine is CD. Uenco 



Ay_CD 



A»~CT{ 



Now it is clear that as C is brought nearer and nearer to 13, the 



figure BUD approaches more and more nearly to the figure of a triangle 



similar to BOF ; and therefore, tho ratio — ;— approaches more and 



more nearly to tl 



llonco wlicn C so moving is jiist 

 of things call BC </,r, and CD dy) 



rfw^OF__ 

 dx OB' 



not an ap- 



•The reader should very carefully note tliM 

 proximate result, but exact. In all tlieso ■ i^ viri,- nrnitg are 

 dealt with, wo are compelled to consider niii . i : - in order 



to learn the nature of tho final state el i '' "in- result 



refers to (ftnt state of things, and not to :ni_. i ih nriliaii. state, 

 linirm-er near. The reader has niisfcd the cKseni inl [loint of the 

 method of limits if ho fails to see this. We have not to deal with 

 approximate resuJtj at all in thus applying tho method of limits. 

 I'crliaiis the beginner may recognise this truth more clearly if J 

 apply tho method to solve a well-known problem. Let it bo required 

 to detcnnino the angle ABT, included between a radius AB of the 

 circle liCF (Fig. 2), and the tangent BT at B. Take O a point near 



B, and draw the secant CBD. Than conceive that C approaches E, 

 carrying the secant along with it. Ohvionnly when C has 

 thns moved np to B, tho Kccant will occupy tho position 

 of the tangent BT. Now in any antecedent position, as C in tho 

 figure, wo have tho triangle ABC isosoolos; and tho equal angles 

 ABC, ACB together, differ from two right angles by the 

 angle A. Ilenco ABC falls short of a right angle by half tho 

 angle A, so that ABD (which together with ABC makes up two 

 right angles) exceeds a right angle by half the angle A. Now 

 when C moves up to B, tho angle A diminishes, and ultimately 

 vanishes. Ilence the difference between ABD and a right ongle 

 ultimately vanishes j so that when the secant CBD has become the 



