【正文】 Spanier, The Fractional Calculus, Academic Press, 1974.　[12]T. J. Osler, Fractional derivatives and the Leibniz rule, Amer. Math. Monthly, 78 (1971), 645–649.　　[13]E. L. Post, Generalized differentiation, Trans. Amer. Math. Soc., 32 (1930) 723–781.　　　[14]B. Ross, editor, Proceedings of the International Conference on Fractional Calculusand its applications, SpringerVerlag, 1975.　　　[15]N. Wheeler, Construction and Physical Application of the fractional Calculus, notes for a Reed College Physics Seminar, 1997.　原文： 　A Child’s Garden of Fractional Derivatives　Marcia Kleinz and Thomas J. Osler　The College Mathematics Journal, March 2000, Volume 31, Number 2, pp. 82–88　　Marcia Kleinz is an instructor of mathematics at Rowan University. Marcia is married and has two children aged four and eight. She would rather research the fractional calculus than clean, and preparing lectures is preferable to doing laundry. Her hobbies include reading, music, and physical fitness.　　Tom Osler (osler) is a professor of mathematics at Rowan University. He received his . from the Courant Institute at New York University in 1970 and is the author of twentythree mathematical papers. In addition to teaching university mathematics for the past thirtyeight years, Tom has a passion for long distance running. Included in his over 1600 races are wins in three national championships in the late sixties at distances from 25 kilometers to 50 miles. He is the author of two running books.　　Introduction　　We are all familiar with the idea of derivatives. The usual notation　or ， or 　　　is easily understood. We are also familiar with properties like　　　　　　But what would be the meaning of notation like or ？Most readers will not have encountered a derivative of “order ” before, because almost none of the familiar textbooks mention it. Yet the notion was discussed briefly as early as the eighteenth century by Leibnitz. Other giants of the past including L’Hospital, Euler, Lagrange, Laplace, Riemann, Fourier, Liouville, and others at least toyed with the idea. Today a vast literature exists on this subject called the “fractional calculus.” Two text books on the subject at the graduate level have appeared recently, [9] and [11]. Also, two collections of papers delivered at conferences are found in [7] and [14]. A set of very readable seminar notes has been prepared by Wheeler [15], but these have not beenpublished.　It is the purpose of this paper to introduce the fractional calculus in a gentle manner. Rather than the usual definition—lemma—theorem approach, we explore the idea of a fractional derivative by first looking at examples of familiar nth order derivatives like and then replacing the natural number n by other numbers like In this way, like detectives, we will try to see what mathematical structure might be hidden in the idea. We will avoid a formal definition of the fractional derivative until we have first explored the possibility of various approaches to the notion. (For a quick look at formal definitions see the excellent expository paper by Miller [8].)　　As the exploration continues, we will at times ask the reader to ponder certain questions. The answers to these questions are found in the last section of this paper. So just what is a fractional derivative? Let us see. . . .　　Fractional derivatives of exponential functions　We will begin by examining the derivatives of the exponential function because the patterns they develop lend themselves to easy exploration. We are familiar with the expressions for the derivatives of ., and, in general, when n is an integer. Could we replace n by 1/2 and write Why not try? Why not go further and let n be an irrational number like or a plex number like1+i ?　　We will be bold and write　 (1)　for any value of , integer, rational, irrational, or plex. It is interes