FROM ELEMRNTAKV ALGEHKA Tl) THE CALCULUS. 4II 



Again, when wc come to velocity (rate of change of dis- 

 placement), the opportunity of driving home the intrinsic limit 

 idea in the word ' rate ' is too often dcliheratelv rejected bv some 

 such subterfuge as " the distance that would be traversed if 

 the velocity remained constant" (the very thing it does not do). 

 The geometrical progression to infinity is another excellent illus- 

 tration ready to hand. 



Having, 1 hope, made out a sufficiently strong case for 

 seizing instead of burking every opportunity of Sfetting the, 

 idea of limits established, I proceed to the details of the calculus. 



I suggest using the words " rate of increase " of a function 

 at first rather than " differential coefficient." The sense in which 

 the rate of increase of .r- is 2.v is grasped easily enough; and 

 I find it needs very little drilling in other easy cases to establish 

 the fundamental proposition that " the rate of increase of / [x) 



is the limit of ^^-^ when .r' ^ .r" as a natural consequence 



.r — ,r 



of simple ideas rather than the isolated but definite straw that 

 some of us used to grasp in the dizzy whirlpool of " differential 

 coefficients with regard to x." 



The only continuous functions thai at this stage have come 

 within the student's experience are included in 



.r" ; sin.v, sin^'.v ; tan.v, tan^'.v ; a'; 

 /(.v) + (/)(.r); /.v.</).v: and/(<^.v). 



It seems strange that it has been so long the custom to 

 import infinite series, with all their difficulties and pitfalls, into 

 these fundamental differentiations. 



It will probably be wise, in teaching, to consider first ,v' : but 



• we proceed to the general case. If n= + - (integers), 



.J 



writing x ' for .v, we have 



7. 

 I. Rate of increase of x ' - 



= , dividing out (x'— X ), 



Lt, 



X + X X + . . . + X '-^ ix 



!— I 



J— I J— 1 ■ V J~' 



X +x X+...+ X ■'■-» 



X- 



■Hx"-\ 



If /i= — , we have 

 J 



X — X ~' — ^ 



L^- X^-X-- - Lt. 



Lt. 



c'(x''_x^) 



i— I i—2 



X ' + X ' x -1- . . . -f X '-' 



X ''■ x '■( X ■'-' + X '-^ X + . . . + X '-f) 



X '"' ; 



-• X -^^-^ 



