arXiv:1801.09553v2 [math.GM] 20 Aug 2018 farto nsc ocpin d conception, limit a the such represent In to ratio. derivative a the calculus of for of therefore deal revised and great was a mathematics, caused within infinitesimal difficulty an of concept The close infinitely as of result zero. own, to thought their likely were on (a But, infinitesimals reals these functions). the continuous smooth, within for be result the potentially other, each could with small examining ratio into infinitely as put these were quantities of When thought quantities. small originally infinitely was calculus The variable. each in differen- infinitesi- differences of (possibly mal) ratio small a arbitrarily as representing notation, expressed tials, Leibniz is the compet- derivative is the A derivative notation where the a notation. for Lagrange had notation ing modern even the and derivative to the Newton, similar of Isaac concept by original interpretations. invented the and was history, fluxion notations rich The long, competing a many had has with variations of calculus The Introduction 1 units cases. of independent number as larger much differentials a treat for for allow to will derivatives ability nota- higher the revised and done second a the often using for is However, tion it derivative. reasons first practical the acting for for unit though generally holistic limit, is a a as It as than rather frac- differentials abuse. a of as and derivative tion the use treat to of problematic units history considered algebraic long a independent have as differentials Treating Abstract elyidpnetuis u,we lcdi ratio in placed when but, units, independent really xedn h leri aiuaiiyo Differentials of Manipulability Algebraic the Extending 1 h lt nttt,[email protected] Institute, Blyth The oahnBartlett Jonathan 2 nttt fMcais A fArmenia of NAS Mechanics, of Institute x n d and y r not are uut2,2018 21, August 1 n strZ.Khurshudyan Zh. Asatur and 1 rte sfatost etetda rcin.I pre- natu- It their fractions. since mistakes, as making treated from be students vents to are which fractions items as allows written it both as for practitioners, calculus and an students with to working leads in This simplification notation overall revised. the is if derivatives expanded higher-order greatly for be can manipulability differentials algebraic of the shown, be will as However, in- of the treatment algebraic improve more differentials. to a dividual effort for allow little algebraically. to been treated notation be has not there should such Therefore, they as reinforced that simply them idea has treating and the concern, when great caused arise not issues units, has that algebraic fact independent the not are assumed generally differentials is it that devel- Because of notation. lack the unfortunate of an opment to led a has dichotomy of This limits apart. represent taken only be cannot they which if ratio they as but available, manipulated differentials written are distinct generally were are there if equations as that means Essen- this it. tially, of interpretation intellectual favored the been u h Langrangian the but rlypeerdteLinzntto,weed where gen- notation, Leibniz infinitesimals the of nota- validity preferred erally the in favoring differen- preferences Those by of tion. paralleled status somewhat ontological was the tials over question This na neetn uno vns ntelt 9hcentury, 19th late the Leibniz in the events, of turn interesting an In unit. the holistic where a notation, is Lagrange deriva-derivative units, the the prefer individual of generally conception as tive limit the represented favoring visually those while least at are to preferring notion, limit as many the However, d ratio with view that pleased smaller. of and not smaller were limit get the changes represent the other, each with x n d and notation y sdsic ahmtclobjects. mathematical distinct as o h eiaielreywnout, won largely derivative the for conception 2 ftedrvtv has derivative the of x n d and y ral inclination is to treat differentials as fractions.1 Ad- Say that it is later discovered that x is a function of t ditionally, there are several little-known but extremely so that x = t2. The problem here is that the chain rule helpful formulas which are straightforwardly deducible for the second derivative is not the same as what would from this new notation. be implied by the algebraic representation.

Even absent these practical concerns, we find that Here we arrive at one of the major problematic points reconceptualizing differentials in terms of algebraically- for using the current notation of the second derivative manipulable terms is an interesting project in its own algebraically. To demonstrate the problem explicitly, if right, and perhaps may help us see the derivative in a one were to take the second derivative seriously as a set 2 new way, and adapt it to new uses in the future. There d y of algebraic units, one should be able to multiply dx2 may also be additional formulas which can in the future dx2 by dt2 to get the second derivative of y with respect to be more directly connected to the algebraic formulation t. However, this does not work. If the differentials are of the derivative. dx2 being treated as algebraic units, then dt2 is the same as dx 2 dt , which is just the first derivative of x with respect to t squared. The first derivative of x with respect to
dx = 2 The Problem of Manipulating t is dt 2t. Therefore, treating the second derivative algebraically would imply that all that is needed to do Differentials Algebraically to convert the second derivative of y with respect to x into the second derivative of y with respect to t is to multiply by 2t 2. When dealing with the first derivative, there are gen- ( ) erally few practical problems in treating differentials dy However, this reasoning leads to the false conclusion algebraically. If y is a function of x, then is the first 2 dx d y = 4 derivative of y with respect to x. This can generally be that dt2 24t . If, instead, the substitution is done treated as a fraction. at the beginning, it can be easily seen that the result should be 30t4: d x 3 For instance, since dy is the first derivative of x with re- y = x spect to y, it is easy to see that these values are merely = 2 the inverse of each other. The inverse function theo- x t dx = 1 = 2 3 rem of calculus states that dy dy . The generaliza- y t dx ( ) tion of this theorem into the multivariable domain es- y = t6 sentially provides for fraction-like behavior within the 5 y′ = 6t first derivative. 4 y′′ = 30t Likewise, in preparation for integration, both sides of the equation can be multiplied by dx. Even in multi- This is also shown by the true chain rule for the sec- variate equations, differentials can essentially be mul- ond derivative, based on Fa`adi Bruno’s formula [2]. tiplied and divided freely, as long as the manipulations This formula says that the chain rule for the second are dealing with the first derivative. derivative should be: 2 d2 y d2 y dx dy d2 x Even the chain rule goes along with this. Let x depend = + (1) dy dt2 dx2 dt dx dt2 on parameter u. If one has the derivative du and mul-   du tiplies it by the derivative dx then the result will be This, however, is extremely unintuitive, and essentially dy makes a mockery out of the concept of using the differ- dx . This is identical to the chain rule in Lagrangian notation. ential as an algebraic unit.

It is well recognized that problems occur when if one It is generally assumed that this is a problem for the tries to extend this technique to the second derivative idea that second differentials should be treated as alge- [1]. Take for a simple example the function y = x3. braic units. However, it is possible that the real prob- dy = 2 lem is that the notation for second differentials has not The first derivative is dx 3x . The second derivative d2y = been given as careful attention as it should. is dx2 6x.

1 The habits of mind that have come from this have even Since many in the engineering disciplines are not formally trained mathematicians, this also can prevent professionals in affected nonstandard analysis, where, despite their ap- applied fields from making similar mistakes. preciation for the algebraic properties of differentials,

2 have left the algebraic nature of the second derivative dx as separate steps. Originally, in the Leibnizian con- either unexamined (as in [3]) or examined poorly (i.e., ception of the differential, one did not even bother solv- leaving out the problematic nature of the second deriva- ing for derivatives, as they made little sense from the tive, as in [4, pg. 4]). original geometric construction of them [5, pgs. 8, 59].

For a simple example, the differential of x3 can be found using a basic differential operator such that d x3 = 2 ( ) 3 A Few Notes on Differential 3x dx. The derivative is simply the differential divided d x3 = 2 Notation by dx. This would yield d( x ) 3x . For implicit derivatives, separating out taking the dif- Most calculus students glaze over the notation for ferential and finding a particular derivative greatly sim- higher derivatives, and few if any books bother to give plifies the process. Given an function (say, z2 = sin q ), ( ) any reasons behind what the notation means. It is im- the differential can be applied to both sides just like any portant to go back and consider why the notation is other algebraic manipulation: what it is, and what the pieces are supposed to repre- 2 = sent. z sin q 2 ( ) d z = d sin q In modern calculus, the derivative is always taken with ( ) ( ( )) 2z dz = cos q dq respect to some variable. However, this is not strictly ( ) required, as the differential operation can be used in a From there, the equation can be manipulated to solve differ- dz dq context-free manner. The processes of taking a for dq or dz , or it can just be left as-is. ential and solving for a derivative (i.e., some ratio of differentials) can be separated out into logically sepa- The basic differential of a variable is normally written rate operations.2 simply as d x = dx. In fact, dx can be viewed merely a shorthand( for) d x . d ( ) In such an operation, instead of doing dx (taking the derivative with respect to the variable x), one would The second differential is merely the differential oper- separate out performing the differential and dividing by ator applied twice [5, pg. 17]: 2 The idea that finding a differential (i.e., similar to a deriva- d d x = d dx = d2 x (2) tive, but not being with respect to any particular variable) can be ( ( )) ( ) separated from the operation of finding a derivative (i.e., differ- Therefore, the second differential of a function is merely entiating with respect to some particular variable) is considered the differential operator applied twice. However, one an anathema to some, but this concept can be inferred directly from the activity of treating derivatives as fractions of differen- must be careful when doing this, as the product rule tials. The rules for taking a differential are identical to those for affects products of differentials as well. taking an implicit derivative, but simply leaving out dividing the final differential by the differential of the independent variable. For instance, d 3x2 dx will be found using the product For those uncomfortable with taking a differential without =( 2 ) = a derivative (i.e., without specifying an independent variable), rule, where u 3x and v dx. In other words: imagine the differential operator d as combining the operations 2 2 2 () d 3x dx = 3x d dx + d 3x dx of taking an implicit derivative with respect to a non-present ( ) ( ( )) ( ) variable (such as q) followed by a multiplication by the differ- = 3x2 d2 x + 6x dx dx ential of that variable (i.e., dq in this example). So, taking the 2 2 2 differential of ex is written as d ex and the result of this opera- = 3x d x + 6x dx tion is ex dx. This is the same as( if) we had taken the derivative with respect to the non-present variable q and then multiplied d2y The point of all of this is to realize that the notation 2 by dq. So, for instance, taking the differential of the function dx ex , the operation would start out with a derivative with respect is not some arbitrary arrangement of symbols, but has a d d to q ex = ex x followed by a multiplication by dq, yielding deep (if, as will be shown, slightly incorrect or mislead- dq ( ) dq just ex dx. ing) meaning. The notation means that the equation Doing this yields the standard set of differential rules, but al- is showing the ratio of the second differential of y (i.e., lows them to be applied separately from (and prior to) a full d d y ) to the square of dx (i.e., dx2).3 derivative. Also note that because they have no dependency on ( ( )) any variable present in the equation, the rules work in the single- variable and multi-variable case. Solving for a derivative is then In other words, starting with y, then applying the dif- merely solving for a ratio of differentials that arise after perform- ferential operator twice, and then dividing by dx twice, ing the differential. It unifies explicit and implicit differentiation into a unified process that is easier to teach, use, and under- 3 In Leibniz notation, dx2 is equivalent to dx 2. If the differ- stand, and requires few if any special cases, save the standard ential of x2 was wanted, it would be written as( d) x2 . The rules requirements of continuity and smoothness. are given in [5, pg. 24]. ( )

3 d2y d2 x arrives at the result dx2 . Unfortunately, that is not the in (3) is that the ratio dx2 reduces to zero. However, same sequence of steps that happens when two deriva- this is not necessarily true. The concern is that, since dx d2 x tives are performed, and thus it leads to a faulty for- dx is always 1 (i.e., a constant), then dx2 should be mulation of the second derivative. zero. The problem with this concern is that we are no d2 x dx longer taking dx2 to be the derivative of dx . Using the dx notation in (3), the derivative of dx would be:

4 Extending the Second Deriva- d d x 2 2 dx = d x dx d x tive’s Algebraic Manipulabil- 2 2 (5) d x
dx − dx dx ity dx In this case, since dx reduces to 1, the expression is ob- d2 x viously zero. However, in (5), the term dx2 is not itself As a matter of fact, order of operations is very impor- necessarily zero, since it is not the second derivative of tant when doing derivatives. When doing a derivative, x with respect to x. one first takes the differential and then divides by dx. The second derivative is the derivative of the first, so the next differential occurs after the first derivative is complete, and the process finishes by dividing by dx 5 The Notation for the Higher again. Order Derivatives However, what does it look like to take the differential of the first derivative? Basic calculus rules tell us that The notation for the third and higher derivatives can the quotient rule should be used: be found using the same techniques as for the second derivative. To find the third derivative of y with respect dy = dx d dy dy d dx d ( ( )) − 2 ( ( )) x dx dx to , one starts with the second derivative and takes the   2 ( ) 2 differential: = dx d y dy d x −2 d dx d y 2 2 dx dx d y dy d x d = dx  dx2 − dx2 2 2 © ª = dx d y dy d x ­ 2 ® 2 ­= d y® dy d x dx dx − dx dx d 2 2 2 2 « dx¬ − dx dx d y dy d x   = dx d2 y dy d2 x dx − dx dx = d − dx3 Then, for the second step, this can be divided by dx,   dx3 d dx d2 y dy d2 x dx d2 y dy d2 x d dx3 yielding: = ( )( ( − ))−( − )( ( )) dy x3 2 d 2 2 d dx d y dy d x 3 3 2 (2 ) 2 2 = (3) d y dy d x d x d y dy d x dx  dx2 − dx dx2 = 3 + 3 ( ) dx2 − dx dx2 − dx2 dx dx dx3 This, in fact, yields a notation for the second derivative which is equally algebraically manipulable as the first Finally, this result is divided by dx: derivative. It is not very pretty or compact, but it d d y works algebraically. dx d d ( x ) 3 2 2 2 ! d y dy d3 x d2 x d y dy d x = 3 3 3 2 2 + 3 ( 4) (6) The chain rule for the second derivative fits this al- dx dx − dx dx − dx dx dx dx gebraic notation correctly, provided we replace each instance of the second derivative with its full form This expression includes a lot of terms not normally (cf. (1)): seen, so some explanation is worthwhile. In this ex- 2 2 2 d y dy d2 x = d y dy d2 x dx 2 + dy d2 x dx d2t pression, d x represents the second differential of x, or d 2 d d 2 d 2 d d 2 d d d 2 d d 2 (4) t − t x x − x x t x t − t t d d x . Therefore, d2 x 2 represents d d x 2. Like-     ( ( )) 4 ( ) 4 ( ( ( ))) This in fact works out perfectly
algebraically. wise, dx represents d x . ( ( )) One objection that has been given to the present au- Because the expanded notation for the second and thors by early reviewers about the formula presented higher derivatives is much more verbose than the first

4 derivative, it is often useful to adopt a slight modifi- this on the second derivative: cation of Arbogast’s D notation (see [6, pgs. 209,218– 2 2 2 = d y dy d x 219]) for the total derivative instead of writing it as Dx y 2 2 4 dx − dx dx algebraic differentials: 3 2 3 2 3 2 dx = d y dx dy d x dx Dx y 3 2 3 2 3 2 2 dy dx dy − dx dx dy 2 = d y dy d x 3 2 2 Dx y 2 2 (7) 2 dx = d y dx d x dx − dx dx Dx y 2 2 3 3 2 2 2 2 dy dy dy − dy d y dy d x d x d y dy d x 3 = +  3 Dx y 3 3 3 2 2 3 ( 4) (8) dx d2 x dx d2 y dx − dx dx − dx dx dx dx D2 y = − x dy dy2 − dy dy2   This gets even more important as the number of deriva- 3 1 d2 x dx d2 y tives increases. Each one is more unwieldy than the D2 y = − x dy dy2 − dy dy2 previous one. However, each level can be interconverted dx ! 3 into differential notation as follows: 1 d2 x dx d2 y D2 y = − x D1 y dy2 − dy dy2 Dn 1 y  x  n = d x− Dx y ( ) (9) It can be seen that this final equation is the derivative dx of x with respect to y. Therefore, it can generally be y x The advantages of Arbogast’s notation over Lagrangian stated that the second derivative of with respect to x notation are that (1) this modification of Arbogast’s can be transformed into the second derivative of with y notation clearly specifies both the top and bottom dif- respect to with the following formula: ferential, and (2) for very high order derivatives, La- 3 2 1 = 2 grangian notation takes up n superscript spaces to write Dx y 1 Dy x (10) − Dx y for the nth derivative, while Arbogast’s notation only   To see this formula in action on a simple equation, con- takes up log n spaces. ( ) sider y = x3. Performing two derivatives gives us: Therefore, when a compact representation of higher or- y = x3 (11) der derivatives is needed, this paper will use Arbogast’s 1 2 5 D y = 3x (12) notation for its clarity and succinctness. x 2 = Dx y 6x (13) 2 According to (10), Dy x (or, x′′ in Lagrangian notation) can be found by performing the following: 3 6 Swapping the Independent 1 D2 x = 6x y −( ) 3x2 and Dependent Variables   6x = − 27x6 In fact, just as the algebraic manipulation of the first 2 5 = − x− (14) derivative can be used to convert the derivative of y 9 with respect to x into the derivative of x with respect to This can be checked by taking successive derivatives of y, combining it with Arbogast’s notation for the second the inverse function of (11): derivative can be used to generate the formula for doing 1 x = y 3

4 1 1 2 The difference between this notation and that of Arbogast is D x = y −3 that we are subscripting the D with the variable with which the y 3 derivative is being taken with respect to. Additionally, we are 2 2 5 = −3 always supplying in the superscript the number of derivatives we Dy x y (15) are taking. Therefore, where Arbogast would write simply D, −9 1 this notation would be written as Dx . 5 (15) can be seen to be equivalent to (14) by substituting It may be surprising to find a paper on the algebraic no- for y using (11): tation of differentials using a non-algebraic notation. The goal, however, is to only use ratios when they act as ratios. When 2 2 3 5 D x = x −3 writing a ratio that works like a ratio is too cumbersome, we y −9( ) prefer simply avoiding the ratio notation altogether, to prevent 2 5 making unwarranted leaps based on notation that may mislead = x− (16) the intuition. −9

5 This is the same result achieved by using the inversion the real exact solutions of which is 3 formula (cf. (10)). 6√2c1 y x = ( ) − 3 3 2 162 x + c2 + 23328c + 162 x + c2 ( ) 1 [ ( )] r q 7 Using the Inversion Formula 3 3 2 162 x + c2 + 23328c1 + 162 x + c2 for the Second Derivative r ( ) [ ( )] q 3 − 3√2 (19) While the inversion formula (cf. (10)) is not original, c1 c2 it is a tool that many mathematicians are unaware of, Here and are integration constants that must be and is rarely considered for solving higher-order differ- determined from given boundary or Cauchy conditions. entials.6 On the other hand, (18) results in As an example of how to apply (10), consider second y3 x y = + c1 y + c2, order ordinary nonlinear differential equations of the ( ) 6 form the real inverse of which exactly coincides with (19). F y′′, y′, y = 0. ( ) Equations of this form can be solved implicitly for 8 Relationship to Historic Leib- F a, b, c = a b3 f c ( ) − ( ) nizian Thought for generic function f . Indeed, consider the equation The view of differentials presented by Leibniz and those 3 2 dy following in his footsteps differed significantly from the D y = f y . (17) x ( ) dx modern-day view of calculus. The modern view of cal-   culus focuses on functions, which have defined inde- Then, by virtue of (10) we derive pendent and dependent variables. The Leibniz view, however, according to [5], is a much more geometric 2 D x = f y . view. There is no preferred independent or dependent y − ( ) variable. Integration of this equation with respect to y twice will provide with The modern concept of the derivative generally implies a dependent and in independent variable. The numer- ator is the dependent variable and the denominator is x y = f y dy dy. (18) ( ) − ( ) the independent variable. In the geometric view, how- ¹ ¹ ever, there are only relationships, and these relation- ships do not necessarily have an implied dependency For simplicity, let relationship. f y = y, ( ) Therefore, Leibnizian differentiation doesn’t occur with so that (17) is reduced to respect to any independent variable. There is no pre- 3 ferred independent variable. Likewise, as we have seen dy D2 y = y , in Sections 6 and 7, the version of the differential pre- x dx   sented here allows for the reversal of variable depen- dency relationships. Similarly, the procedure of differ- 6 The authors of this paper, as well as several early reviewers, entiation given in Section 3 which allows us to formu- had originally thought that the inversion formula was a new find- ing. Again, that is the usefulness of the notation. Specific formu- late the new notation for the second derivative given in las such as the inversion formula do not need to be taught, as they (3) follows the Leibnizian methodology, where the dif- simply flow naturally out of the notation. Even though the inver- ferentiation is done mechanically without considering sion formula is not new with this paper, showing how the present variable dependencies. authors were able to use it to good benefit demonstrates the ben- efit of an improved notation—practitioners needs not memorize endless formulas, but they can be developed straightforwardly as Leibniz did, however, consider certain kinds of variables needed based upon basic intuitions. which map very directly to what we would consider as

6 “independent” variables. In the Leibniz conception, However, if we assume that x is truly the independent what we would consider an “independent” variable is variable, then this means that d2 x = 0 and therefore dy d2 x a variable whose first derivative is considered constant. the whole expression dx dx2 reduces to 0 as well. This This leads to numerous simplifications of differentials d2y reduces (3) to the modern notation of dx2 . Addition- because, if a differential is constant, by standard differ- ally, if we take the assumption that x is the independent ential rules its differential is zero. Therefore, if x is the variable, then the problems identified in Section 2 dis- independent variable (using modern terminology) then appear, because x, as an independent variable, cannot that implies that dx is constant. If dx is a constant then be dependent on t.8 (even if it is an infinitely small, unknowable constant), then that means that its differential is zero. There- d2y In addition to (3) being reducible to d 2 under the as- fore, d2 x and higher differentials of x reduce to zero, x 7 sumption that x is the independent variable, the Leib- simplifying the equation. nizian view also gives a set of tools that allows us to d2y reinflate instances of 2 into (3). Euler showed that, As an example, given the equation dx given an equation from a specific “progression of vari- xy = 3 ables” (i.e., a particular choice of an independent vari- able), we can modify that equation in order to see what the first differential of this would be given by it would have been if no choice of independent vari- able had been made. According to [5, pg. 75], the sub- + = x dy y dx 0 stitution for reinflating a differential from a particular progression of variables (i.e., a particular independent and the second differential of this would be given by variable) into one that is independent of the progres- x d2 y + 2 dx dy + y d2 x = 0. sion of variables (i.e., no independent variable chosen), an expansion practically identical to (3) can be used. Then, you could simplify the equation by choosing any single differential to hold constant. This is referred to in Leibnizian thought as choosing a “progression of variables,” and it is identical to choosing an indepen- 9 Future Work dent variable [5, pg. 71]. Therefore, if one chooses x as the independent variable, then dx is constant, and The notation presented here provides for a vast im- 2 = therefore d x 0. Thus, the equation reduces to provement in the ability for higher order differentials 2 to be manipulated algebraically. x d y + 2 dx dy = 0. This improved notation yields several potential areas However, if y is the independent variable, then dy is 2 for study. These include: held constant and therefore its differential, d y = 0. This leads to the equation 1. developing a general formula for the algebraic ex- + 2 = 2 dx dy y d x 0. pansion of higher derivatives, This understanding explains the success of the modern 2. identifying additional second order differential notation of the second derivative. The notation given equations that are solvable by swapping the de- in (3) is pendent and independent variable, d2 y dy d2 x D2 = . x dx2 − dx dx2 3. finding other ways that differential equations can be rendered solvable using insights from the new 7As a way of understanding this, imagine the common inde- pendent variable used in physics, especially prior to relativity— notation, time. Especially consider the way that time flows in a pre- relativistic era. It flows in a continual, constant fashion. There- 4. finding further reductions in special formulas that fore, if the flow of time (i.e., dt) is constant, then by the rules of can be rendered by using algebraically manipulable differentiation the second differential of time must be zero. Thus, notations, an independent variable is one which acts in a similar fashion to time. Another way to consider this is to consider the indepen- 8To be clear, there is nothing preventing someone from mak- dence of the independent variable. It’s changes (i.e., differences) ing an independent variable dependent on a parameter. However, are, by definition, independent of anything else. Therefore, we doing so then brings them around to needing to use the form of may not assign a rule about the differences between the values. the second derivative defined here (which does not presume a Thus, because there is no valid rule, the second differential may particular choice of independent variable), or a compensating not be zero, but it is at most undefinable by definition. mechanism such as Fa`adi Bruno’s formula.

7 5. investigating the conjecture that the second differ- ential of independent variables are always zero and its potential implications, and 6. extending this project to allow partial differentials to be algebraically manipulable.

10 Acknowledgements

The authors would like to thank Aleks Kleyn, Chris Burba, Daniel Lichtblau, George Monta˜nez, and others who read early versions of this manuscript and provided important feedback and suggestions.

References

[1] E. W. Swokowski, Calculus with Analytic Geome- try. PWS Publishers, alternate ed., 1983. [2] W. P. Johnson, “The curious history of Fa‘a di Bruno’s formula,” American Mathematical Monthly, pp. 217–234, 2002. [3] J. M. Henle and E. M. Kleinberg, Infinitesimal Cal- culus. Dover Publications, 1979. [4] H. J. Keisler, Elementary Calculus: An Infinitesi- mal Approach. PWS Publishers, second ed., 1985. [5] H. J. M. Bos, “Differentials, higher-order differen- tials, and the derivative in the Leibnizian calculus,” Archive for History of Exact Sciences, vol. 14, no. 1, pp. 1–90, 1974. [6] F. Cajori, A History of Mathematical Notations Volume II. Open Court Publishing, 1929.

8