By Brian H. Chirgwin, Charles Plumpton

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Q— x (P + g) (Pg - ^2) (P2-z2)(g2-*2)' 44 A COURSE OF MATHEMATICS If y == [f(x)]g(x\ we can only find ay/ax tiation. / = g{%) - l o g / O r ) . 1 dw /, * , ,, . flfia;)·/'^) y v / ö / v =g'(x)-\ogf(x) + ^ y dz * ' ' / ( s') * r . / = ax, then log y = x log a. Examples, (i) 1 dy d(a*) — —— = log a and therefore —-— = ax log a. y ax ax If y = xx, then (ii) log y = x log x ; 1 di/ — _ = l+log*, ά(«*) . - . +log*). Hyperbolic functions. There exist certain combinations of t h e exponential functions ex a n d e~x with properties which bear a close formal analogy with those of t h e trigonometric functions.

Cosh (x + y) = cosh x cosh 2/ + sinh # sinh 2/. Similarly sinh (a; + y) = sinh # cosh z/ + cosh # sinh 2/. A hyperbolic i d e n t i t y m a y be formally obtained from a trigonometric i d e n t i t y simply b y changing 'sin' into 'sinh' a n d 'cos' into 'cosh' except FIG. 10. The graphs of sinh x, cosh x, and tanh x. t h a t wherever the product of t w o sines occurs in t h e trigonometric identity a negative sign m u s t be introduced in t h e hyperbolic identity. 6). 46 A C O U R S E OF M A T H E M A T I C S Examples.

Since e~x = l/e*, it m u s t be positive for x < 0 also. F u r t h e r , e° = 1 and lim e~ x — lim 1/e* = 0 . The graph of ex is shown in Fig. 9. - Le* 4 3 2 ^ -3 -2 -1 A 0 -2 ^ ^ 2 3 f FIG. 9. The graphs of e* and l££ a;. Examples, (i) D[eK*>] = /'(a) e/<*>. (ii) If p is an integer, lim xpe~x = lim 1+^ + ^ + + (Ρ + Ι)ϊ" 1 = lim = 0 a;-f» -far*> + i 2! + (P+I)! whatever the value of p . A similar result holds if p is not an integer; the proof requires only slight modification. This limit indicates that, for large values of x9 ex is of a higher order of magnitude than any power of x.