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Extra info for A Treatise on Linear Differential Equations, Volume I: Equations with Uniform Coefficients
Xlxl a) J(x) b) J(x) = c) J(x) = Ixl~ d) J(x) = lxi! I for x i 0 for x = 0 Let J : D --+ IR (D C IR) be a function, x E D. a) Show that J is differentiable at x precisely if the following limits both exist and coincide: e) 2) 29 J(x) = o J~(x):= lim e-+z JW - J(x) , f~(x):= ~ - x eED,e
In particular (i), and thereby the above considerations, hold for every z E D. _ In (3) we let m tend to 00 and obtain from (2) IfN(X) - f(x) - (lN(Z) - f(z»1 ~ Ix - zl'f/· (4) For N, which depends only on 'f/, and x we find a 8 > 0 with IfN(X) - fN(Z) - (x - z)ffv(x) I ~ 'f/lx - zl for Ix - zl < 8. 2). It follows from (2), (4) and (5) that Ix - zl < 8. Since this holds for every xE D and for all z with Ix - zl < 8, it follows from If(x) - f(z) - g(x)(x - z)1 ~ 3'f/lx - zl, if our characterization of differentiability, that f'(x) exists and I'(x) = g(x).
Satisfies f'(x) = f(x) 6) More generally, f(x) =e C "', for all x E IR. cE IR, satisfies f'(x) = ce c", = cf(x) for all x E IR, as follows from 1), 5) and the chain rule. 7 yields in conjunction with 5) log'( x ) = 8) 1 1 = -~--:exp' (log x) exp (log x) 1 x for x > 0. 8 Let f : D -+ IR be a differentiable function. We say that f is twice differentiable at x E D if the derivative f' of f is differentiable at x. ,,(x) := (f')'(x) (the derivative of f' at x), and also ~ (x) in place of f" (x). (The latter symbol is explained as (~ ) f = (d~)2f = d~(*)') Inductively, we say that a k-times differentiable function f : D -+ IR is (k + l)-times differentiable at x E D if the kth derivative f(k) = ~ fis differentiable at x.