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By Nicolaas Govert de Bruijn

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If we set Sf^^ g, then formula F enables us to compute F'. S(f -f^ ~ f -Sf-L - S(Df -S^). While it is immaterial which ant i derivative of f^ we use in the expression on the right side, It is essential that on both places the same antiderlvatlve Sf^ Is selected. Example: S logabs ~ S(logabs-l) ~ S(logabs-DJ) ^» J«logabs - 8(J»rec) ^ J*logabs - Sl~*J*logabs - J. 45 III. ON FUNCTIONS OP HIGHER RANK 1, The Algebra of Functions of Higher Rank. As In the first chapter, we shall denote functions by small letters f,g,h, ...

J)-D(J-J) • £«rec(J«J)»2«J *rec(J»J)»J =(rec J«rec J)«J *(rec*rec)*J » rec*(rec*J) * rec*l « rec. Next we compute D abs. We have D abs »p(exp log abs) *D(exp logabs) *D exp logabs«D logabs * exp logabs • rec * abs »rec «sgn. We remark that the formulae D log " reo and D abs • sgn by virtue of postulate III entail the formula D logabs * rec. Applying the last formula and postulate III we obtain D( logabs f) » D logabs f «Df " rec f *Df. 4. Logarithmic and Exponential Derivation. The formulae at 'the end of the two preceding sections can also be written as follows: 36 Df • f«D(logabs f) and Df • rec exp f*D(exp f).

D tan 0 * 1. For this function we have Q D tan • 1 4- tan m Q rec cos . In the classical analysis, for each constant a the function tan (a-x) satisfies postulate 1. The function associating tan x with z is the only one which satisfies postulates 1 and 2. In a paper "e and ir In Elementary Calculus11 (to appear in the near future ) we describe how the postulates D tan 0*1 and D exp 0 * 1 in conjunction with the functional equations for the tangential and exponential functions yield an intuitive introduction of ir and e, as well as a simple development of the ^natural* tangential and exponential functions ex and tan x (x measured in radians).

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