Sept. 1, 1882. 



KNOAVLEDGE • 



237 



(^ir iKatOnnatiral Column* 



EASY LESSONS IN THE DIFFERENTIAL CALCULUS. 



No. IX. 



Bt Bichabd a. Peoctor. 



TWO functions still remain to be dealt with, log .r and a', after 

 which the student will be able to differentiate any functions 

 whatever, simple or complex. 



First, then, let !/ — log,, T. Increasing ,t to x+ ^x, whereby y 

 is increased to y + A y, and then subtracting, we have 

 A!/=loga {x+ t.x)-\og„x 



=loga 11 + -) writing h for A x, for convenience. 



=_j_n-l!+i^_&c.l 



log.aLo! 2a? Sx' J 



1 r^-A*z_ici 



Ay 



' Now when h is vei-y small, all the terms within brackets, except 

 the fii-st,"may be neglected, for thev are less in absolute value than 

 h_r, . h.h' 

 in? 



■&C.1 



a quantity vanishing with h. Hence, finally, 



when Aw, Ax are both indefinilely small, we have 

 dy_ 1 

 da! itlog,.a 

 It follows that if !/ = log,,i 



Next let V = 

 as follows : — 



conteit ourselves by proccedii 

 1 



^ = a' logoff. 



[There is in reality no objection to regarding -1. as the reciprocal 

 dx 



of — , however true it may le that — :- is to be regarded as a 



dy dx 



single expression.] 



It follows that if !/ = e' 



^^=e'also. 

 doc 

 We can now differentiate any expression we please, however 

 complex. There is scarcely ever any room for much ingenuity in 

 the choice of methods in differentiating. Still at times we find 

 that space is saved, and we avoid the chance of errors in writing, 

 by modifying onr method of procedure. 



Here are a few examples of differentiation : — 



1. Let y = x (a' + u') ^a'-x' 

 Here, proceeding on the straightforward course, we get 



= (a' + x'). 



'-x' + 2x>s 



■y'a 



But we might have proceeded thus : — 



log y = log X + log (a' + ,t') + J log (a' - x') 

 ,', differentiating with respect to x 

 I dti \ . 2x X 



<'-4x 



dy _a' + a'a)'— 4^' 



y-Iog (log j) 

 dy 1 1 



y = lo? Oog [log']) 



dy^l J^ 1 



dx xlogi' log (logx) 



y^^^l'. Here we treat x' as the vaiiable exponent. 



= loga. a'' (x x^'+j:'. log r) 

 = loga.a'' x^ (1-tlog r). 

 y = a''"S' 

 dy _\og a . a^'<"' 



d7 J ■ 



y-.v-'' 



g = log...^'x'(l + :og.r) +x'(x^) 



= log ,v . 1 •' .I-' (1 + log t) + x^ ,r^' 



I have occasion for a hearty laugh at a mistake of my own, of a 

 full size — nay, overgrown. Gulliver tells us that the tailors of 

 Laputa measured folk for their clothes by mathematical methods — 

 observing altitudes, triangulating, and so forth, but that generally 

 some mistake in the computation caused the clothes to be exceed- 

 ingly ill-fitting. I made the captain of a racing eight calculate by 

 the differential calcnlns the proper size for one of his crew, but a 

 very bad blunder made him obtain a wrong result. And really the 

 blunder was so bad that the wonder is he did not deduce by the 

 differential calculus a man two feet or twenty feet high, which 

 might have involved him in difficulty. An odd thing is that my 

 blunder was made eleven years ago, the problem having originally 

 appeared in my short series of Easy Lessons in the English ilechanic 

 (which lessons must have interested many, seeing that no one noted 

 the mistake ; thirteen readers of Knowledge have already pointed 

 it out). In our columns the problem was simply reprinted and 

 corrected from the printed original : (I think >n additional error 

 crept in ; but have no means of kno%ving, being away from my 

 back volumes of the Evytith Mechanic.) The solution should, of 

 course, have run thus : — 



Let AM (breadth of man) = 2j ; man's weight = mx' ; moment 

 of his weight around centre line of boat's length = mx'. GB = 



mx'{l-x) = mx'-vix*. Therefore ^ =3mx'-4n«i'; and equating 



to zero, X = -, or AM = g : so that since a man 1 ft. broad weighs, 

 according to our assumption, G stone, our new oarsman should weigh 

 6 (2) = 8i = 20J stone. 



I need hardly say that if, eleven . ears ago, n y solution had been 

 correctly obtained, with this ridiculous result, I should havo 

 slightly modified the conditions. For instance, I might have 

 assumed that a one-foot seat corresponded to a weight of 4 stone 

 instead of 6, in which case the reasonable weight ISJ stone would 

 have resulted for the new oarsman. 



The value of the differential calculus is not affected by my 

 mistake, which, however, teaches the value of care, especially in 

 dealing with very easy problems. 



A. W. Bawtrce, one of the thirteen who note the above mistake, 

 points out the following clerical errors in the Easy Lessons (I am 

 only able to check his crrrcctions for articles in Part 10, being 

 away from home, and not having Part 9 by mo) : — 



p. laC, 1. 14 fi-om bottom, 1,000 should be 10,000. 

 1. 13 „ „ y should be x. 



p. 1.j4', 1. 8, tan I and tan (j- -h :Sj) should be cot x and cot (x + Ax), 



p. 155', 1. 2, ^ should bo ilf 

 dy dx 



)). 171', bottom line, " functions of: " should be" functions of Jr.'' 



J). 1S8', 1. 18 from bottom, the equation should bo 



£•' ^ 



dx°2(«' + a')* 

 p. 20-1', I. 21 from bottom, 5 = iic., should be y= Ac. 

 Jloat of these are obvious misprints, but they should be corrected 

 in the text. 



