New Approach for Fractional Order Derivatives: Fundamentals and Analytic Properties

mathematics

Article New Approach for Fractional Order Derivatives: Fundamentals and Analytic Properties

Ali Karcı

Department of Computer Engineering, Faculty of Engineering, Inönü˙ University, 44280 Malatya, Turkey; [email protected] or [email protected]; Tel.: +90-422-3774841; Fax: +90-422-3410046

Academic Editor: Hari M. Srivastava Received: 25 February 2016; Accepted: 21 April 2016; Published: 4 May 2016

Abstract: The rate of change of any function versus its independent variables was defined as a derivative. The fundamentals of the derivative concept were constructed by Newton and l’Hôpital. The followers of Newton and l’Hôpital defined fractional order derivative concepts. We express the derivative defined by Newton and l’Hôpital as an ordinary derivative, and there are also fractional order derivatives. So, the derivative concept was handled in this paper, and a new definition for derivative based on indefinite limit and l’Hôpital’s rule was expressed. This new approach illustrated that a derivative operator may be non-linear. Based on this idea, the asymptotic behaviors of functions were analyzed and it was observed that the rates of changes of any function attain maximum value at inflection points in the positive direction and minimum value (negative) at inflection points in the negative direction. This case brought out the fact that the derivative operator does not have to be linear; it may be non-linear. Another important result of this paper is the relationships between complex numbers and derivative concepts, since both concepts have directions and magnitudes.

Keywords: derivatives; fractional calculus; fractional order derivatives

1. Introduction The asymptotic behaviors of functions can be analyzed by velocities or rates of change in functions, while very small changes occur in the independent variables. The concept of rate of change in any function versus change in the independent variables was defined as a derivative, and this concept attracted many scientists and mathematicians such as Newton, l’Hôpital, Leibniz, Abel, Euler, Riemann, etc. Isaac Newton defined the fundamentals of classical mechanics and this study contains rates of changes of functions [1]. He collected his works in Philosophiæ Naturalis Principia Mathematica, a book that includes geometrical proofs, gravitational force law, and attraction of bodies [1]. He was the first scientist who concerned himself with the concept of derivatives/fluxions. On the other hand, Newton tried to determine the change in length of distance in terms of the velocity of bodies, and the change in velocity of bodies in terms of acceleration. L’Hôpital was a follower of Newton in that he defined the concept of a derivative and generalized this concept. There are other mathematicians who dealt with the concepts of derivatives/fluxions. The first important and detailed work in differential calculus and differential geometry was done by l’Hôpital [2,3]. L’Hôpital generalized the ideas of Newton through variations on calculus. Leibniz is another scientist who expounded on differential calculus and infinitesimal calculus [4]; he mastered the mathematics of his day and developed his own calculus over the short span of a few years [4]. Newton, l’Hôpital, and Leibniz are not the only mathematicians who dealt with variations of calculus. Some mathematicians tried to explain the ratio between the two displacements of at least two variables [5], since it is important for analyzing the asymptotic behaviors of functions.

Mathematics 2016, 4, 30; doi:10.3390/math4020030 www.mdpi.com/journal/mathematics Mathematics 2016, 4, 30 2 of 15

The asymptotic behaviors of functions can be regarded as the rates of displacements of functions versus rates of displacement of independent variables—in other words, the rates of movement of functions. The term fluxion indicates motion and the idea of the fluxional calculus developed from the concept that a geometrical magnitude was the result of continuous motion of a point, line, or plane [6]. This motion, speaking of plane curves, could be considered in reference to coordinate axes as the result of two motions, one in the direction of the X-axis and the other in the direction of the Y-axis [6]. The velocity of the X-component and the Y-component were called “fluxions” by Newton [1,6]. The velocity of a point is represented by an equation involving the fluxions x and y [6]. The problems of variation of calculus are attractive to mathematicians and there are several familiar mathematicians who focused their attention on these problems such as Newton, l’Hôpital, Leibniz, Euler, Abel, Caputo, Riemann, Grünwald, Miller, Ross, et al. [7–11]. The problems of variation of calculus and infinitesimal calculus are not solved completely, and there are still open problems in fractional variation of calculus. There are a lot of studies about the fractional variations of calculus [11–22]. Euler, Caputo, Riemann, Abel, et al. dealt with fractional variations of calculus and fractional order calculus and systems. Karci defined the fractional order derivative concept in a different way by using indefinite limits and the l’Hôpital rule [23–25]. Some popular fractional order derivative methods, such as Euler, Caputo, and Riemann-Liouville, can be summarized as follows. dn xm Γpm`1q m´ n The Euler method is dxn “ Γpmń`1q x and its deficiencies can be illustrated for constant and identity functions. Assume that f (x) = cx0 where c is a constant and c P R. Assume that n = 1 and 1 α “ 4 and: n m 1 0 5 d x Γpm ` 1q mń d 4 x Γ ´ 1 “ x “ “ 4 cx 4 dxn Γpm ´ n ` 1q 1 3 dx 4 Γ ` 4 ˘ The derivative of any constant function is always zero; however,` ˘ the result of fractional order 2 derivative with respect to the Euler method is different from zero. Assume that f (x) = x, n = 1 and α “ 3 .

2 1 d 3 x Γp1 ` 1q 2 Γp2q 1 1´ 3 3 2 “ 2 x “ x ‰ 1 3 Γ 1 ´ ` 1 4 dx 3 Γ 3 ` ˘ ´ ¯ n t α 1 d f pvqdv The Riemann-Liouville method is for function f (t) aDt f ptq “ Γpn´αq dt α´n`1 . The ra pt´vq fractional order derivative also can be applied to constant and identity functions´ ¯ with respect to the Riemann-Liouville method. 0 2 Assuming that f (x) = cx , c P R, n = 1 and α “ 3 ,

t t n p q 2 Dα f ptq “ 1 d f v dv “ D 3 f ptq “ 1 d cdv “ c ´ 1 ‰ 0 a t Γpn´αq dt α´n`1 a t Γ 1 dt 2 Γ 1 2 ra pt´vq p 3 q ra pt´vq 3 p 3 q pt´aq 3 ´ ¯ ˆ ˙ The obtained result is inconsistent, since the result is a function of x. However, the initial function is a constant function and its derivative is zero, since there is no change in the dependent variable. The same case is valid for identity functions.

n t 2 t t Dα f ptq “ 1 d f pvqdv “ D 3 f ptq “ 1 d xdx “ 1 d xdx a t Γpn´αq dt αń`1 a t 1 dt 2 ´1`1 1 dt 2 ra pt´vq Γp 3 q ra pt´xq 3 Γp 3 q a pt´xq 3 ´ ¯ 2 4 1 3 9 3 ş “ 1 3apt ´ aq ` 4 pt ´ aq ‰ 1 Γp 3 q ´ ¯ t pnq C α 1 f pvqdv The Caputo method is a Dt f ptq “ Γpαńq α`1ń . The Caputo method does not have ra pt´vq inconsistency for constant functions; however, it has inconsistency for identity functions. Assuming 2 that f (x) = x, n = 1 and α “ 3 , Mathematics 2016, 4, 30 3 of 15

n 111ddxdxdxdxtttfvdv() 2  3 atDft() n 122 at Dft () 11 n dtaaa()tv 11 dt  dt ()tx33  () tx 33 

1924 33 3(at a ) ( t a ) 1 1 4  3

1 t fvdv()n () The Caputo method is C Dft () . The Caputo method does not have at  1 n Mathematics 2016, 4, 30 ()n a ()tv 3 of 15 inconsistency for constant functions; however, it has inconsistency for identity functions. Assuming 2 that f(x) = x, n = 1 and   , 3

t tt()n t 1 C  111pnq fvdv() dv Dft1() f pvqdv 1 dv  3(1 t a )3 1 1 C α at 12 n 3 a Dt f ptq “ ()n ()tv “ 1111 “ 3pt ´ aq ‰ 1 α aaα`1ń 1 3 2 `1´1 1 Γp ´ nq pt ´ vq Γ´ ()tvpt ´ vq 3  Γ ´ ża 333ża  3 ´ ¯ ´ ¯ ´ ¯ Due to these deficiencies, deficiencies, there is a need for a new approach to fractional order derivatives; this paper contains such a definitiondefinition and some importantimportant propertiesproperties ofof thisthis approach.approach. This paper is organized as follows. The The motivation motivation of of this this paper paper will will be be presented presented in in Section Section 22.. Section 33 illustrates illustrates the the applications applications of of rational/irrational rational/irrational ordersorders ofof derivatives.derivatives. SectionSection4 4is is the the definitiondefinition andand detailsdetails ofof the the new new approach approach. and Section puts forth 5 puts the forth analytical the analytical results of results this new of approachthis new approachfor the derivative for the derivative concept. Finally, concept. the Finally, paper isthe concluded paper is concluded in Section5 in. Section 6.

2. Motivation Motivation The rate of change of functionsfunctions is anan importantimportant conceptconcept toto examineexamine inin mathematics.mathematics. For this purpose, the concept of derivatives was identified, identified, because the rate of change of the function gives detailed information about a system modeled by that function, and the nature of the problems or systems. For this purpose, an athlete’s speed may be examined on a ski-jumpski‐jump ramp (Figure1 1).). TheThe trajectory of movement of an athlete on the ski-jumpski‐jump ramp can be considered as a curve. The rate of change of the athlete’s speed increases until the inflection inflection point (Figure 1 1);); after after thatthat pointpoint thethe raterate ofof change of the athlete’s speed will decrease. So, the rate of change has its maximum at the inflectioninflection point. The rate of change of the athlete’s speed will be zero at point E (the local extremum point). The rate of change can be determined as shown in Figure 11,, wherewhere itit cancan bebe seenseen thatthat thethe raterate ofof change change has magnitude and direction. The The rate rate of of change as as seen in Figure 11 isis directeddirected toto positive,positive, andand thethe situation of ratesrates isis seenseen in in Figure Figure1 .1. This This concept concept and and the the relationships relationships with with complex complex numbers numbers will will be bediscussed discussed in detailin detail in subsequentin subsequent sections sections of thisof this paper. paper.

FigureFigure 1. 1. TheThe movement movement of of an an athlete athlete on on the the ski ski-jump‐jump rampramp and and the the rates rates of of change. change.

In order to make this situation more clear and understandable, trigonometric and polynomial functions can be used. To this end, sine and cosine and two polynomial functions can be examined as examples. The rate of change can be regarded as the velocity of function change. In order to determine the behaviors of rate of change for any function, there will be very small increments/decrements in the independent variable (these increments/decrements are equal) and the response of the dependent variable to these small increments/decrements must be examined. Mathematics 2016, 4, 30 4 of 15

Assume that f (x) = sin(x) and x P r1, 2πs. This closed interval can be divided into four closed intervals as follows:

π π 3π I “ rInf , E s “ 0, I “ rE , Inf s “ , π I “ rInf , E s “ π, and 1 1 1 2 2 1 2 2 3 2 2 2 ” ı ” ı „  3π I “ rE , Inf s “ , 2π 4 2 3 2 „ 

The first interval I1 = [Inf1, E1] can be examined for equal-length increments/decrements in the independent variable x such as ∆x1 = ∆x2 = ... = ∆xn-1 = ∆xn = ∆x << 1. The first point is inflection point [Inf1, E1]; assume that yinf1 = sin(xinf1) is valid. (a) xinf1 P rinf1, E1s and 1 ď i ď n

x1inf1 = xinf1 + ∆x y1inf1 = yinf1 + ∆y1inf1 = sin(xinf1 + ∆x) x2inf1 = x1inf1 + ∆x = xinf1 + 2∆x y2inf1 = y1inf1 + ∆y2inf1 = yinf1 + ∆y1inf1 + ∆y2inf1 = sin(x1inf1 + ∆x) = sin(xinf1 + 2∆x) x3inf1 = x2inf1 + ∆x = x1inf1 + 2∆x = xinf1 + 3∆x y3inf1 = y2inf1 + ∆y3inf1 = y1inf1 + ∆y2inf1 + ∆y3inf1 = sin(x2inf1 + ∆x) = sin(x1inf1 + 2∆x) = sin(xinf1 + 3∆x) ...... n xninf1 = x(n-1)inf1 + ∆x = xinf1 + ∆x “ xinf1 ` n∆ x i“1 n ř yninf1 “ sin xinf1 ` ∆x “ sin pxinf1 ` n∆xq. ˆ i“1 ˙ ř At this point, the changes in the dependent variable y = f (x) are ∆y1inf1, ∆y2inf1, ... , ∆y(n-1)inf1, ∆yninf1 and inequalities for these changes are as follows: ∆y1inf1 ě ∆y2inf1 ě ... . ě ∆y(n-1)inf1 ě ∆yninf1 and |∆y1inf1| ě |∆y2inf1| ě ... . ě |∆y(n-1)inf1| ě |∆yninf1|. So, the velocities of change can be identiﬁed as follows:

∆y ´ ∆y ∆y ´ ∆y ∆ypn´2qinf1 ´ ∆ypn´1qinf1 ∆ypn´1qinf1 ´ ∆yninf1 1inf1 2inf1 ě 2inf1 3inf1 ě .... ě ě ě 0 ∆x ∆x ∆x ∆x (b) The same argument can be made for the second closed interval ∆x P I2 “ rE1, Inf2s,

and y = f (x) = sin(xE1). x1E1 = xE1 + ∆x y1E1 = yE1 + ∆y1E1 = sin(xE1 + ∆x) x2E1 = x1E1 + ∆x = xE1 + 2∆x y2E1 = y1E1 + ∆y2E1 = yE1 + ∆y1E1 + ∆y2E1 = sin(x1E1 + ∆x) = sin(xE1 + 2∆x) x3E1 = x2E1 + ∆x3 = x1E1 + 2∆x = xE1 + 3∆x y3E1 = y2E1 + ∆y3E1 = y1E1 + ∆y2E1 + ∆y3E1 = sin(x2E1 + ∆x) = sin(x1E1 + 2∆x) = sin(xE1 + 3∆x) ...... n xnE1 = x(n-1)E1 + ∆x = xE1 + ∆x “ xE1 ` n∆x i“1 n ř ynE1 “ sin xE1 ` ∆x “ sin pxE1 ` n∆xq. ˆ i“1 ˙ ř At this point, the changes in the dependent variable y = f (x) are ∆y1E1, ∆y2E1, ... , ∆y(n-1)E1, ∆ynE1 and inequalities for these changes are as follows: ∆y1E1 ě ∆y2E1 ě ... . ě ∆y(n-1)E1 ě ∆ynE1 and |∆y1E1| ď |∆y2E1| ď ... ď |∆y(n-1)E1| ď |∆ynE1|. So, the velocities of change can be identiﬁed as follows:

∆y ´ ∆y ∆y ´ ∆y ∆ypn´2qE1 ´ ∆ypn´1qE1 ∆ypn´1qE1 ´ ∆ynE1 1E1 2E1 ě 2E1 3E1 ě .... ě ě ě 0 ∆x ∆x ∆x ∆x Mathematics 2016, 4, 30 5 of 15

(c) The same argument can be done for the second closed interval ∆x P I3 “ rInf2, E2s, and y = f (x) = sin(xinf2). x1inf2 = xinf2 + ∆x y1inf2 = yinf2 + ∆y1inf2 = sin(xinf2 + ∆x) x2inf2 = x1inf2 + ∆x2 = xinf2 + 2∆x y2inf2 = y1inf2 + ∆y2inf2 = yinf2 + ∆y1inf2 + ∆y2inf2 = sin(x1inf2 + ∆x) = sin(xinf2 + 2∆x) x3inf2 = x2inf2 + ∆x = x1inf2 + 2∆x = xinf2 + 3∆x y3inf2 = y2inf2 + ∆y3inf2 = y1inf2 + ∆y2inf2 + ∆y3inf2 = sin(x2inf2 + ∆x) = sin(x1inf2 + 2∆x) = sin(xinf2 + 3∆x) ...... n xninf2 = x(n-1)inf2 + ∆x = xinf2 + ∆x “ xinf2 ` n∆x i“1 n ř yninf2 “ sin xinf2 ` ∆x “ sin pxinf2 ` n∆xq. ˆ i“1 ˙ ř At this point, the changes in the dependent variable y = f (x) are ∆y1inf2, ∆y2inf2, ... , ∆y(n-1)inf2, ∆yninf2 and inequalities for these changes are as follows: ∆y1inf2 ď ∆y2inf2 ď ... ď ∆y(n-1)inf2 ď ∆yninf2 and |∆y1inf2| ě |∆y2inf2| ě ... ě |∆y(n-1)inf2| ě |∆yninf2|. So, the velocities of change can be identiﬁed as follows:

∆y ´ ∆y ∆y ´ ∆y ∆ypn´2qinf2 ´ ∆ypn´1qinf2 ∆ypn´1qinf2 ´ ∆yninf2 1inf2 2inf2 ď 2inf2 3inf2 ď .... ď ď ď 0 ∆x ∆x ∆x ∆x (d) The same argument can be made for the second closed interval ∆x P I4 “ rE2, Inf2s, and y = f (x) = sin(xE2).

x1E2 = xE2 + ∆x y1E2 = yE2 + ∆y1E2 = sin(xE2 + ∆x) x2E2 = x1E2 + ∆x = xE2 + 2∆x y2E2 = y1E2 + ∆y2E2 = yE2 + ∆y1E2 + ∆y2E2 = sin(x1E2 + ∆x) = sin(xE2 + 2∆x) x3E2 = x2E2 + ∆x = x1E2 + 2∆x = xE2 + 3∆x y3E2 = y2E2 + ∆y3E2 = y1E2 + ∆y2E2 + ∆y3E2 = sin(x2E2 + ∆x) = sin(x1E2 + 2∆x) = sin(xE2 + 3∆x) ...... n xnE2 = x(n-1)E2 + ∆x = xE2 + ∆x “ xE2 ` n∆x i“1 n ř ynE2 “ sin xE2 ` ∆x “ sin pxE2 ` n∆xq. ˆ i“1 ˙ ř At this point, the changes in the dependent variable y = f (x) are ∆y1E2, ∆y2E2, ... , ∆y(n-1)E2, ∆ynE2 and inequalities for these changes are as follows: ∆y1E2 ě ∆y2E2 ě ... ě ∆y(n-1)E2 ě ∆ynE2 and |∆y1E2| ě |∆y2E2| ě ... ě |∆y(n-1)E2| ě |∆ynE2|. So, the velocities of change can be identiﬁed as follows:

∆y ´ ∆y ∆y ´ ∆y ∆ypn´2qE2 ´ ∆ypn´1qE2 ∆ypn´1qE2 ´ ∆ynE2 1E2 2E2 ě 2E2 3E2 ě .... ě ě ě 0 ∆x ∆x ∆x ∆x Similar arguments can be made for one period of a cosine function; this period was divided into four closed intervals as follows:

π π 3π I “ rE , Inf s “ 0, I “ rInf , E s “ , π I “ rE , Inf s “ π, and 1 1 1 2 2 1 2 2 3 2 2 2 ” ı ” ı „  3π I “ rInf , E s “ , 2π 4 2 3 2 „ 

The very small increments/decrements in independent variable x can be ∆x1 = ∆x2 = ... = ∆xn-1 = ∆xn = ∆x. First of all, the changes in the dependent variable y = f (x) = cos(x) can be examined for the closed interval I1. At this point, the changes in the dependent variable y = f (x) are ∆y1E1, ∆y2E1, Mathematics 2016, 4, 30 6 of 15

... , ∆y(n-1)E1, ∆ynE1 and inequalities for these changes are as follows: ∆y1E1 ě ∆y2E1 ě ... ě ∆y(n-1)E1 ě ∆ynE1 and |∆y1E1| ď |∆y2E1| ď ... ď |∆y(n-1)E1| ď |∆ynE1|. So, the velocities of change can be identiﬁed as follows:

∆y ´ ∆y ∆y ´ ∆y ∆ypn´2qE1 ´ ∆ypn´1qE1 ∆ypn´1qE1 ´ ∆ynE1 1E1 2E1 ď 2E1 3E1 ď .... ď ď ě 0 ∆x ∆x ∆x ∆x

The very small increments/decrements in independent variable x can be ∆x1 = ∆x2 = ... . = ∆xn-1 = ∆xn = ∆x. The changes in the dependent variable y = f (x) = cos(x) can be examined for the closed interval I2. At this point, the changes in the dependent variable y = f (x) are ∆y1inf1, ∆y2inf1, ... , ∆y(n-1)inf1, ∆yninf1 and inequalities for these changes are as follows: ∆y1inf1 ď ∆y2inf1 ď ... ď ∆y(n-1)inf1 ď ∆yninf1 and |∆y1inf1| ě |∆y2inf1| ě ... ě |∆y(n-1)inf1| ě |∆yninf1|. So, the velocities of change can be identiﬁed as follows:

∆y ´ ∆y ∆y ´ ∆y ∆ypn´2qinf1 ´ ∆ypn´1qinf1 ∆ypn´1qinf1 ´ ∆yninf1 1inf1 2inf1 ď 2inf1 3inf1 ď .... ď ď ď 0 ∆x ∆x ∆x ∆x

The changes in the dependent variable y = f (x) = cos(x) can be examined for the closed interval I3. At this point, the changes in the dependent variable y = f (x) are ∆y1E2, ∆y2E2, ... ., ∆y(n-1)E2, ď ∆ynE2 and inequalities for these changes are as follows: ∆y1E2 ď ∆y2E2 ď ... ď ∆y(n-1)E2 ď ∆ynE2 and |∆y1E2| ď |∆y2E2| ď ... ď |∆y(n-1)E2| ď |∆ynE2|. So, the velocities of change can be identiﬁed as follows:

∆y ´ ∆y ∆y ´ ∆y ∆ypn´2qE2 ´ ∆ypn´1qE2 ∆ypn´1qE2 ´ ∆ynE2 1E2 2E2 ď 2E2 3E2 ď .... ď ď ď 0 ∆x ∆x ∆x ∆x

The changes in the dependent variable y = f (x) = cos(x) can be examined for the closed interval I4. At this point, the changes in the dependent variable y = f (x) are ∆y1inf2, ∆y2inf2, ... , ∆y(n-1)inf2, ∆yninf2 and inequalities for these changes are as follows: ∆y1inf2 ě ∆y2inf2 ě ... ě ∆y(n-1)inf2 ě ∆yninf2 and |∆y1inf2| ě |∆y2inf2| ě ... ě |∆y(n-1)inf2| ě |∆yninf2|. So, the velocities of change can be identiﬁed as follows:

∆y ´ ∆y ∆y ´ ∆y ∆ypn´2qinf2 ´ ∆ypn´1qinf2 ∆ypn´1qinf2 ´ ∆yninf2 1inf2 2inf2 ě 2inf2 3inf2 ě .... ě ě ě 0 ∆x ∆x ∆x ∆x

3. Applications of Rational/Irrational Orders The theoretical information given in Section2 can be verified by applications of trigonometric and polynomial functions. To this end, sine, cosine, and increasing and decreasing polynomial functions are selected. These functions are sin(x) for x P r0, 2πs, cos(x) for x P r0, 2πs, x3 ´ 7x2 for x P r´1000, 1000s and ´x3 ´ 7x2 for x P r´1000, 1000s. Figures2 and3 show the rates of changes for sine functions and the obtained results support the claims of Figure1. The red circles in both figures depict the inflection points. Figures4 and5 depict the same cases for the cosine function. In the case of Figures3 and5 while power is equal to 1, the rate of change is the same as an ordinary derivative. The domains of sine and cosine functions were determined as a period of functions. The inflection points for sine function are {(0, 0), (0, π), (0, 2π)} and large changes occur at these points for sine function (Figures2 and3). The inflection points for cosine function are {(π/2, 0), (3π/2, 0} and large changes occur at these points for cosine function (Figures4 and5). Mathematics 2016, 4, 30 7 of 15 Mathematics 2016, 4, 30 7 of 15 Mathematics 2016, 4, 30 7 of 15

Rational Powers of Velocity of Change of Function sin(x) Rational Powers of Velocity of Change of Function sin(x) Mathematics 2016, 4, 30 f(x)=sin(x) Order=0 7 of 15 f(x)=sin(x) 1 20 Order=0 1 20 Rational Powers of Velocity of Change of Function sin(x) 0.5 10 0.5 10 f(x)=sin(x) Order=0 0 1 200 0 0 -0.50.5 10 -0.5 -10 -10 -1 0 0 -1 -20 0 pi/2 pi 3*pi/2 2*pi -20 0 pi/2 pi 3*pi/2 2*pi -0.50 pi/2 pi 3*pi/2 2*pi -10 0 pi/2 pi 3*pi/2 2*pi Order=1/5 Order=2/5 20-1 Order=1/5 20 Order=2/5 20 -2020 0 pi/2 pi 3*pi/2 2*pi 0 pi/2 pi 3*pi/2 2*pi 10 10 10 Order=1/5 10 Order=2/5 20 20 0 0 0 0 10 10 -10 -10 -10 -10 0 0 -20 -20 -20 0 pi/2 pi 3*pi/2 2*pi -20 0 pi/2 pi 3*pi/2 2*pi -100 pi/2 pi 3*pi/2 2*pi -10 0 pi/2 pi 3*pi/2 2*pi

-20 -20 Figure 2. The rational0 orderspi/2 ofpi rates3*pi/2 of change2*pi for the0 sinepi/2 functionpi 3*pi/2 (Orders2*pi are 0, 1/5, 2/5). FigureFigure 2. 2.The The rational rational orders orders of of rates rates of of change change forfor thethe sine function (Orders (Orders are are 0, 0, 1/5, 1/5, 2/5). 2/5). Rational Powers of Velocity of Change of Function sin(x) Figure 2. The rational ordersRational ofPowers rates of of Velocity change of Change for the of sineFunction function sin(x) (Orders are 0, 1/5, 2/5). f(x)=sin(x) Order=3/5 Order=3/5 1 f(x)=sin(x)Rational Powers of Velocity of Change10 of Function sin(x) 1 10 Order=3/5 f(x)=sin(x) 5 0.5 10 0.5 1 5 0 0 00.5 50 -0.5 0 -50 -0.5 -5 -1 -1-0.5 -10-5 0 pi/2 pi 3*pi/2 2*pi -10 0 pi/2 pi 3*pi/2 2*pi 0 pi/2 pi 3*pi/2 2*pi 0 pi/2 pi 3*pi/2 2*pi -1 -10 Order=1, or Ordinary Derivative 0 pi/2Order=4/5pi 3*pi/2 2*pi 0Order=1,pi/2 or Ordinarypi 3*pi/2 Derivative2*pi 10 Order=4/5 10 1 Order=4/5 1 Order=1, or Ordinary Derivative 510 0.51 5 0.5 0 5 0.50 0 0 -5 0 -0.50 -5 -0.5 -5 -0.5-1 -10 -1 -10 0 pi/2 pi 3*pi/2 2*pi 0 pi/2 pi 3*pi/2 2*pi -1 -100 pi/2 pi 3*pi/2 2*pi 0 pi/2 pi 3*pi/2 2*pi 0 pi/2 pi 3*pi/2 2*pi 0 pi/2 pi 3*pi/2 2*pi Figure 3. The rational orders of rates of change for the sine function (Orders are 3/5, 4/5, 1). Figure 3. The rational orders of rates of change for the sine function (Orders are 3/5, 4/5, 1). FigureFigure 3. The 3. The rational rational orders orders of of rates rates of of change change forfor the sine function function (Orders (Orders are are 3/5, 3/5, 4/5, 4/5,1). 1). Rational Powers of Velocity of Change of Function cos(x) Rational Powers of Velocity of Change of Function cos(x) Rational Powers of Velocity of Change of Function cos(x) Order=0 f(x)=cos(x) Order=0 1 f(x)=cos(x) 20 Order=0 1 f(x)=cos(x) 20 1 20 0.5 10 0.50.5 10 0 0 0 0 0 -0.5 -0.5 -10 -0.5 -10 -1 -1 -1 -20 0 pi/2 pi 3*pi/2 2*pi -20 0 pi/2 pi 3*pi/2 2*pi 0 0 pi/2pi/2 pipi 3*pi/23*pi/2 2*pi2*pi 0 pi/2pi/2 pipi 3*pi/23*pi/2 2*pi2*pi Order=1/5 Order=2/5 Order=2/5 20 Order=1/5Order=1/5 20 Order=2/5 2020 20 10 10 1010 10

0 0 00 0 0

-10-10 -10-10 -10 -10 -20-20 -20-20 -20 0 0 pi/2pi/2 pipi 3*pi/23*pi/2 2*pi2*pi -20 00 pi/2pi/2 pipi 3*pi/23*pi/2 2*pi2*pi 0 pi/2 pi 3*pi/2 2*pi 0 pi/2 pi 3*pi/2 2*pi FigureFigureFigure 4. The4. 4.The The rational rational rational orders orders orders of of of rates rates rates of of of change changechange for for thethethe cosine cosine function function (Orders (Orders (Orders are are are 0, 0,1/5, 0, 1/5, 1/5, 2/5). 2/5). 2/5). Figure 4. The rational orders of rates of change for the cosine function (Orders are 0, 1/5, 2/5).

Mathematics 2016, 4, 30 8 of 15 Mathematics 2016, 4, 30 8 of 15

Mathematics 2016, 4, 30 Rational Powers of Velocity of Change of Function cos(x) 8 of 15

Rational Powers of Velocity of Change of Function cos(x)Order=3/5 f(x)=cos(x) 10 1 Order=3/5 f(x)=cos(x) 10 0.51 5

0.50 50

-0.50 -50

-0.5-1 -10-5 0 pi/2 pi 3*pi/2 2*pi 0 pi/2 pi 3*pi/2 2*pi -1 -10 0 pi/2 Order=4/5pi 3*pi/2 2*pi 0Order=1,pi/2 or Ordinarypi 3*pi/2Derivative2*pi 10 1 Order=4/5 Order=1, or Ordinary Derivative 105 0.51

50 0.50

-50 -0.50

-1 -10-5 -0.5 0 pi/2 pi 3*pi/2 2*pi 0 pi/2 pi 3*pi/2 2*pi -1 -10 0 pi/2 pi 3*pi/2 2*pi 0 pi/2 pi 3*pi/2 2*pi FigureFigure 5. 5. TheThe rationalrational ordersorders of of rates rates of of change change for for the the cosine cosine function function (Orders (Orders are are 3/5, 3/5, 4/5,4/5, 1). 1). Figure 5. The rational orders of rates of change for the cosine function (Orders are 3/5, 4/5, 1). 3 2 The samesame ideaidea cancan bebe arguedargued for for polynomial polynomial functions, functions, specifically specifically functions functionsf (fx(x)) = =x x3 ´ − 77xx2 for 3 2 xxP[ r´The 1000,1000]1000, same 1000 ideas and can ff( xbe) =argued −´xx3 − ´ 7for7xx 2 forpolynomialfor x P[ r´ 1000,1000]1000, functions, 1000.s .The specifically The domains domains functions for for these these f (x functions) = x3 − 7 x were2 for selectedx [ 1000,1000] to cover theand inflectioninflection f(x) = −x 3points − 7x2 and for extremum x [ 1000,1000] points. . TheThe The domainsinflection inflection forpoint point these for for functionsff(x(x) )= = xx3 − 3 ´7werex72x is2 3 2 isselected(7/3, (7/3, 1372/27); 1372/27); to cover it is the illustrated it is inflection illustrated by points a red by a circle.and red extremum circle. The inflection The points. inflection point The for point inflection f(x) for = −xf (point − x)7 =x´ isforx 3(− ´f7/3,(x7) x −=2 686/27);xis3 − (´7x7/3,2 isit ´(7/3,is 686/27);also 1372/27); illustrated it is it also is byillustrated illustrated a red circle. by by a The red a red circle.black circle. diamondThe The inflection black points point diamond are extremumfor f( pointsx) = −x points3 are − 7x extremum2 isfor (− both7/3, − pointsfunctions.686/27); for it bothisThe also approach functions. illustrated in The Section by a approach red 2 wascircle. verified in The Section black by2 applications wasdiamond verified points for by polynomials applicationsare extremum as for seenpoints polynomials in forFigures both as 6–9.functions. seen in FiguresThe approach6–9. in Section 2 was verified by applications for polynomials as seen in Figures 6–9. Rational Powers of Velocity of Change of a Polynomial 10 Rational Powers of Velocity of Change of a Polynomial 3 2 f(x)=x -7x Order=0 5 100 10 3 2 0 f(x)=x -7x Order=0 5 10050 50 -50 50 500 -5 -100 0 100 200 0 0 -100 0 100 200 -50 0 -50 -100-50 -20 -10 0 10 20 -50-1000 -500 0 500 1000 -100 Order=1/5 Order=2/5 -20 -10 0 10 20 30 -1000 -500 0 500 1000 50 Order=1/5 Order=2/5 3020 600 50 2010 600400 0 10 -200 0 200 400200 0

-50 -200 0 200 2000 -1000 -500 0 500 1000 -1000 -500 0 500 1000 -50 0 -1000 -500 0 500 1000 Figure 6. The rates of change for f(x) = x3-1000 − 7x2 orders-500 {0,0 1/5,500 2/5}. 1000

3 2 FigureFigure 6. 6.The The rates rates of of change change for forf ( fx()x) = =x x3 ´ − 7x2 ordersorders {0, 1/5, 2/5}. 2/5}.

Mathematics 2016, 4, 30 9 of 15 Mathematics 2016, 4, 30 9 of 15 Mathematics 2016, 4, 30 9 of 15 Mathematics 2016, 4, 30 9 of 15 Rational Powers of Velocity of Change of a Polynomial Rational Powers of Velocity of Change of a Polynomial f(x)=-xRational3-7x 2Powers of Velocity of Change of a PolynomialOrder=3/5 100 f(x)=-x3-7x2 Order=3/5 100 3 2 Order=3/5 f(x)=-x -7x 10000 100 50 10000 8000 50 10000 8000 500 6000 8000 0 6000 4000 0 6000 -50 4000 2000 -50 4000 2000 -100-50 0 2000 -100-20 -10 0 10 20 -10000 -500 0 500 1000 -100-20 -10 0 10 20 -10000 -500 0 500 1000 4 Order=1, or Ordinary Derivative -20x 10 -10 Order=4/50 10 20 -1000Order=1,6 -500 or Ordinary0 Derivative500 1000 4 x 10 x 10 Order=4/5 6 4 3 xOrder=1, 10 or Ordinary Derivative x 10 Order=4/5 3 6 15 x 10 15 3 2 1015 2 10 2 105 1 5 1 5 1 0 0 -10000 -500 0 500 1000 -1000 -500 0 500 1000 -1000 -500 0 500 1000 0 0 -10000 -500 0 500 1000 -1000 -500 0 500 1000 -1000 -500 0 500 1000 Figure 7. The rates of change for f(x) = x33 − 7x2 2orders {3/5, 4/5, 1}. FigureFigure 7. The 7. The rates rates of changeof change for forf ( fx()x) = =x x3 − ´77xx2 ordersorders {3/5, {3/5, 4/5, 4/5, 1}. 1}. Figure 7. The rates of change for f(x) = x3 − 7x2 orders {3/5, 4/5, 1}. Rational Powers of Velocity of Change of a Polynomial Rational Powers of Velocity of Change of a Polynomial Rational Powers of Velocity of Change of a Polynomial 3 2 Order=0 10 f(x)=-x -7x 10 100 3 2 Order=0 f(x)=-x -7x 0 100 3 2 Order=0 10 f(x)=-x -7x 50 0 100 50 50 -100 50 -10 -100 0 100200 50 -100 0 100200 50 -10 0 0 -100 0 100200 0 0 -500 0 -50 -50 -100-50 -50 -100-20 -10 0 10 20 -50-1000 -500 0 500 1000 -20 -10 0 10 20 -1000 -500 0 500 1000 -100 Order=1/5 Order=2/5 -20 -10 Order=1/50 10 20 -1000 -500 0 500 Order=2/51000 50 50 50 Order=1/5 50 200 Order=2/5 200 50 500 0 200 0 -500 100 0 100 -50-1000 0 1000 0 -50-1000 0 1000 100 -1000 0 1000 0 -50 0 -50 0 -50-1000 -500 0 500 1000 -1000 -500 0 500 1000 -1000 -500 0 500 1000 -1000 -500 0 500 1000 -1000Figure-500 8. The0 rates500 of change1000 for f(x) = −-1000x3 − 7x-5002 orders0 {0, 1/5,500 2/5}.1000 3 2 FigureFigure 8. The 8. The rates rates of changeof change for forf ( xf()x =) =´ −xx3 − ´77xx orders2 orders {0, {0, 1/5, 1/5, 2/5}. 2/5}. Figure 8. The rates of change for f(x) = −x3 − 7x2 orders {0, 1/5, 2/5}. Rational Powers of Velocity of Change of a Polynomial Rational Powers of Velocity of Change of a Polynomial Order=3/5 Rational3 Powers2 of Velocity of Change of a Polynomial f(x)=-x -7x 0 Order=3/5 100 f(x)=-x3-7x2 100 3 2 0 Order=3/5 f(x)=-x -7x 0 10050 -1000 50 -1000 50 -1000 0 -2000 0 -2000 0 -2000 -50 -3000 -50 -3000 -100-50 -3000 -100-20 -10 0 10 20 -1000 -500 0 500 1000 -20 -10 0 10 20 -1000 -500 0 500 1000 -100 Order=1, or Ordinary Derivative -20 4 -10 0 10 20 -1000 -500 0 500 1000 x 10 Order=4/5 Order=1,6 or Ordinary Derivative 4 x 10 6 0 x 10 Order=4/5 0 xOrder=1, 10 or Ordinary Derivative 0 4 0 6 x 10 Order=4/5 x 10 0 0 -5 -1 -5 -1 -5 -1 -10 -2 -10 -2 -10 -2 -15 -3 -15-1000 -500 0 500 1000 -1000-3 -500 0 500 1000 -1000 -500 0 500 1000 -15 -1000-3 -500 0 500 1000 -1000 -500 0 500 1000 -1000 -500 0 500 1000 Figure 9. The rates of change for f(x) = x3 − 7x2 orders {3/5, 4/5, 1}. Figure 9. The rates of change for f(x) = x3 − 7x2 orders {3/5, 4/5, 1}. 3 2 FigureFigure 9. The 9. The rates rates of of change change for forf ( fx(x)) = = xx3 − ´77xx 2ordersorders {3/5, {3/5, 4/5, 4/5, 1}. 1}.

Mathematics 2016, 4, 30 10 of 15 Mathematics 2016, 4, 30 10 of 15

AA similarsimilar case case can can be be considered considered for for negative negative order, order, and and to thisto this end, end, sine sine function function was was selected selected for illustration.for illustration. Figure Figure 10 illustrates 10 illustrates that inthat the in case the ofcase negative of negative orders, orders, the same the commentssame comments can be can made be formade fractional for fractional order derivatives. order derivatives. The negative The order negative reverses theorder direction reverses of the the increase/decrease, direction of the so theincrease/decrease, inequalities in Section so the inequalities2 can be rephrased. in Section 2 can be rephrased.

FigureFigure 10. 10.The The rates rates of of change change for forf ( fx()x =) = sin( sin(x)x) for for orders orders { ´{−1,1,´ −0.8, −´0.6}.0.6}.

4.4. Analytical Analytical Approach Approach and and Results Results TheThe meaningmeaning ofof derivativederivative isis thethe raterate ofof changechange oror velocityvelocity ofof changechange inin thethe dependentdependent variablevariable versusversus thethe changeschanges inin thethe independentindependent variablesvariables [ [23–25].23–25]. Thus,Thus, thethe derivativederivative ofoff f((xx)) == cx0 is:is: limfx()() h fx lim cc lim f px ` hq ´ f pxq  lim c0´ c hh00()xh x“ h “ 0 h Ñ 0 px ` hq ´ x h Ñ 0 h In the case of an identity function, it is: In the case of an identity function, it is: limfx()() h fx lim ()xh x   1 hh00()xh x () xh  x lim f px ` hq ´ f pxq lim px ` hq ´ x “ “ 1 So, the definition for hrational/irrationalÑ 0 px ` hq ´ orderx derivativeh Ñ 0 p xcan` hbeq ´ consideredx as follows. →  DefinitionSo, the 1. definition f(x): R for R is rational/irrational a function, R orderand the derivative rational/irrational can be considered order derivative as follows. can be considered as follows: Definition 1. f(x): R Ñ R is a function, α P R and the rational/irrational order derivative can be considered lim fxh()() fx as follows: fx() () (1) h  0 α()xh x α pαq lim f px ` hq ´ f pxq f pxq “ α α (1) Before handling applications, the newh definitionsÑ 0 px `forh qrational/irrational´ x order derivatives must be rephrased.Before handling Definition applications, 1 is a classical the new definition definitions of derivative, for rational/irrational and it has an order indefinite derivatives limit must such be rephrased.0 Definition 1 is a classical definition of derivative, and it has an indefinite limit such that 0that , while h = 0. In this case, Definition 1 can be rephrased as seen in Definition 2. 0 , while0 h = 0. In this case, Definition 1 can be rephrased as seen in Definition 2. DefinitionDefinition 2.2. AssumeAssume thatthat f(x):f(x): RR Ñ→R R is is a a function, function,α P  R andand L(.)L(.) isis aa l’Hôpitall’Hôpital process.process. TheThe rational/irrationalrational/irrational orderorder derivativederivative ofof f(x)f(x) is:is:

α α α α df ()() xdp f hpx` fhq´ xf pxqq 1 α´1 pαq lim f px ` hq ´ f pxq lim dh f p1xq f pxq f pxq “ limL fxh()()α fx“ lim α fxfʹ()“ () x (2) fx() h()Ñ0 Lpx ` hq ´ xα h Ñ 0 dhdppx`hq ´xαq xα ´1 (2)   1 hhˆ00()xh x˙ dxh (()  xdh ) x dh The new definition of rational/irrational order derivative (Definition 2) can be applied to some specific functions such as constant, identity, sine, and cosine functions. The velocity of change for a

Mathematics 2016, 4, 30 11 of 15

The new definition of rational/irrational order derivative (Definition 2) can be applied to some specific functions such as constant, identity, sine, and cosine functions. The velocity of change for a constant function is zero whatever the independent variable changes. The velocity of change for an identity function is 1 whatever the order of derivative and change of independent variable. The derivatives of sine, cosine, and polynomial functions for different orders are illustrated in Figures2–9. These results demonstrated that the new definition obeys velocities of change of functions in extremum points, inflection points, etc. The comments and descriptions in Section2 can be supported with analytical results. The velocities of change of any functions can be expressed by derivatives. The point of this study is to analyze the powers of derivative with respect to all real powers. ` Theorem 4.1. Assume that f(x) is any function such that f(x): R Ñ R, and α1, α2, ... , αn P R . Let f(x) be a positive, monotonically increasing/decreasing function in which the following conditions hold:

` (a) If α1, α2, ... , αn P R , α1 ď α2 ď ... ď αn and f(x) is positive and monotonically increasing, then f pα1qpxq ď f pα2qpxq ď ..... ď f pαn´1qpxq ď f pαnqpxq. ` (b) If α1, α2, ... , αn P R , α1 ď α2 ď ... ď αn, and f(x) is positive and monotonically decreasing, then f pα1qpxq ě f pα2qpxq ě ..... ě f pαn´1qpxq ě f pαnqpxq.

Proof.

(a) f (x) is a monotonically increasing function, so f (xi) ď f (xj) for xi ď xj, @xi, xj P R. Then α αj α αj f i pxk ` hq ď f pxk ` hq and f i pxkq ď f pxkq where 1 ď i < j ď n, k is any index, αi, αj are two pα q pα q αj pαjq constants and αi ď αj. This case implies that f i pxk ` hq ´ f i pxkq ď f pxk ` hq ´ f pxkq α αi αi αj j and pxk ` hq ´ xk ď pxk ` hq ´ xk . So, this implies the following quotient:

α α αj αj lim f i pxk ` hq ´ f i pxkq lim f pxk ` hq ´ f pxkq L α α ď L α h Ñ 0 px ` hq i ´ x i h Ñ 0 αj j ˜ k k ¸ ˜ pxk ` hq ´ xk ¸

This implies the following important relations. Step 1. Assuming that n = 2, there are two rational/irrational powers of functions such as α1, ` α2 P R , α1 ď α2. Then:

lim f α1 px ` hq ´ f α1 px q lim f α2 px ` hq ´ f α2 px q L k k ď L k k α1 α1 α2 α2 h Ñ 0 ˜ pxk ` hq ´ xk ¸ h Ñ 0 ˜ pxk ` hq ´ xk ¸

Step 2. Assuming that there are n ´ 1 rational/irrational powers of rates of change, there are ` n ´ 1 rational/irrational powers of functions such as α1, α2, ... , αn´1 P R , α1 ď α2 ... ď αn-1. Then:

α α α α αn´1 αn´1 lim f 1 pxk ` hq ´ f 1 pxkq lim f 2 pxk ` hq ´ f 2 pxkq lim f pxk ` hq ´ f pxkq L α α ď L α α ď ... ď L α h Ñ 0 px ` hq 1 ´ x 1 h Ñ 0 px ` hq 2 ´ x 2 h Ñ 0 αn´1 n´1 ˜ k k ¸ ˜ k k ¸ ˜ pxk ` hq ´ xk ¸

` Step 3. Assuming that α1, α2,..., αn P R , α1 ď α2 ď ... ď αn, the following inequality holds:

αn´1 αn´1 αn αn pn´1q lim f pxk ` hq ´ f pxkq lim f pxk ` hq ´ f pxkq pnq f pxq “ L α ď L α α “ f pxq. h Ñ 0 αn´1 n´1 h Ñ 0 px ` hq n ´ x n ˜ pxk ` hq ´ xk ¸ ˜ k k ¸

This means that f pαn´1qpxq ď f pαnqpxq and f pαiqpxq ď f pαnqpxq for 1 ď i ď n ´ 1.

(b) f (x) is a monotonically decreasing function, so f (xi) ě f (xj) for xi ď xj, @xi, xj P R. Then α αj α αj f i pxk ` hq ě f pxk ` hq and f i pxkq ě f pxkq where 1 ď i < j ď n, k is any index, αi, αj are two Mathematics 2016, 4, 30 12 of 15

pα q pα q αj pαjq constants and αi ď αj. This case implies that f i pxk ` hq ´ f i pxkq ě f pxk ` hq ´ f pxkq α αi αi αj j and pxk ` hq ´ xk ď pxk ` hq ´ xk . So, this implies the following quotient:

α α αj αj lim f i pxk ` hq ´ f i pxkq lim f pxk ` hq ´ f pxkq L α α ě L α h Ñ 0 px ` hq i ´ x i h Ñ 0 αj j ˜ k k ¸ ˜ pxk ` hq ´ xk ¸

This implies the following important relations. Step 1. Assuming that n = 2, there are two rational/irrational powers of functions such as α1, ` α2 P R , α1 ď α2. Then:

lim f α1 px ` hq ´ f α1 px q lim f α2 px ` hq ´ f α2 px q L k k ě L k k α1 α1 α2 α2 h Ñ 0 ˜ pxk ` hq ´ xk ¸ h Ñ 0 ˜ pxk ` hq ´ xk ¸

Step 2. Assuming that there are n ´ 1 rational/irrational powers of rates of changes, ` then there are n ´ 1 rational/irrational powers of functions such as α1, α2, ... , αn´1 P R , α1 ď α2 .... ď αn-1. Then:

` Step 3. Assuming that α1, α2,..., αn P R , α1 ď α2 ď ... ď αn, the following inequality holds:

αn´1 αn´1 αn αn pn´1q lim f pxk ` hq ´ f pxkq lim f pxk ` hq ´ f pxkq pnq f pxq “ L α ď L α α “ f pxq h Ñ 0 αn´1 n´1 h Ñ 0 px ` hq n ´ x n ˜ pxk ` hq ´ xk ¸ ˜ k k ¸

This means that f pαn´1qpxq ď f pαnqpxq and f pαiqpxq ď f pαnqpxqfor 1 ď i ď n ´ 1. ´ Theorem 4.2. Assume that f(x) is any function such that f(x): R Ñ R, and α1, α2, ... , αn P R . Let f(x) be a positive, monotonically increasing/decreasing function in which the following conditions hold:

´ (a) If α1, α2, ... , αn P R , α1 ě α2 ě ... ě αn and f(x) is positive and monotonically increasing, then f pα1qpxq ě f pα2qpxq ě ... ě f pαn´1qpxq ě f pαnqpxq. ´ (b) If α1, α2, ... , αn P R , α1 ě α2 ě ... ě αn, and f(x) is positive and monotonically decreasing, then f pα1qpxq ě f pα2qpxq ě ..... ě f pαn´1qpxq ě f pαnqpxq.

Proof. The proof can be handled in two steps: increasing function and decreasing function.

(a) f (x) is a positive, monotonically increasing function, so f (xi) ď f(xj) for xi ď xj, @xi, xj P R, and α1, ´ α α α α α2 P R and α1 ď α2. Then f 1 px ` hq ě f 2 px ` hq and f 1 pxq ě f 2 pxq, |α1| ě |α2|.

|α | |α | d p f 1 pxq´ f 1 px`hqq α α |α | |α | 1 |α1|`1 lim f 1 pxi ` hq ´ f 1 pxiq dh f 1 pxq f 1 px`hq f pxq pxq A “ L α α “ |α | “ 1 1 x|α1|´px`hq 1 |α1|`1 h Ñ 0 ˜ pxi ` hq ´ xi ¸ d f pxq |α | dh x|α1|px`hq 1

and,

|α | |α | d p f 2 pxq´ f 2 px`hqq α α |α | |α | 1 |α2|`1 lim f 2 pxi ` hq ´ f 2 pxiq dh f 2 pxq f 2 px`hq f pxq pxq B “ L α α “ |α | “ 2 2 x|α2|´px`hq 2 |α2|`1 h Ñ 0 ˜ pxi ` hq ´ xi ¸ d f pxq |α | dh x|α2|px`hq 2 Mathematics 2016, 4, 30 13 of 15

where f |α1|`1pxq ď f |α2|`1pxq and x|α1|`1 ď x|α2|`1. This case implies the following inequality:

f 1pxq pxq|α1|`1 f 1pxq pxq|α2|`1 pxq|α1|`1 pxq|α2|`1 A ě B ñ ě ñ ě f |α1|`1pxq f |α2|`1pxq f |α1|`1pxq f |α2|`1pxq

This case is for two orders, and it can be enlarged to other orders. For α1 ě α2 ě ... ě αn and |α1| ď |α2| ď ... ď |αn|,

f 1pxq pxq|α1|`1 f 1pxq pxq|α2|`1 pxq|αn´1|`1 pxq|αn|`1 ě ě ... ě ě f |α1|`1pxq f |α2|`1pxq f |αn´1|`1pxq f |αn|`1pxq

(b) g(x) is a positive, monotonically decreasing function, so g(xi) ě g(xj) for xi ď xj, @xi, xj P R, and ´ α α α α α1, α2 P R and α1 ď α2. Then g 1 px ` hq ď g 2 px ` hq and g 1 pxq ď g 2 pxq, |α1| ě |α2|. 1 Assuming that f (x) is a positive, monotonically increasing function, gpxq “ f pxq is a positive, monotonically decreasing function:

d |α | |α | α α f 1 px ` hq ´ f 1 pxq 1 |α |´1 lim g 1 pxi ` hq ´ g 1 pxiq dh f pxq f 1 pxq A “ L α α “ |α | “ ´ 1 1 ´ x|α1|´px`hq 1 ¯ 3|α1|´1 h Ñ 0 ˜ pxi ` hq ´ xi ¸ d x |α | dh x|α1|px`hq 1

and,

d |α | |α | α α f 2 px ` hq ´ f 2 pxq 1 |α |´1 lim g 2 pxi ` hq ´ g 2 pxiq dh f pxq f 2 pxq B “ L α α “ |α | “ ´ 2 2 ´ x|α2|´px`hq 2 ¯ 3|α2|´1 h Ñ 0 ˜ pxi ` hq ´ xi ¸ d x |α | dh x|α2|px`hq 2

where f |α2|´1pxq ď f |α1|´1pxq and x3|α2|´1 ď x3|α1|´1. This case implies the following inequality.

f 1pxq f |α1|´1pxq f 1pxq f |α2|´1pxq f 1pxq f |α1|´1pxq f 1pxq f |α2|´1pxq A ď B ñ ´ ě ´ ñ ´ ě ´ x3|α1|´1 x3|α2|´1 x3|α1|´1 x3|α2|´1

This case is for two orders, and it can be enlarged to other orders. For α1 ě α2 ě ... ě αn and |α1| ď |α2| ď ... ď |αn|,

f 1pxq pxq|α1|´1 f 1pxq pxq|α2|´1 pxq|αn´1|´1 pxq|αn|´1 ´ ě ě ... ě ě f |α1|´1pxq f |α2|´1pxq f |αn´1|´1pxq f |αn|´1pxq

Any complex number has direction and magnitude. It is known that derivative has direction and magnitude. So the relationship between complex numbers and derivatives must be veriﬁed. Theorem 4.3. Assuming that f(x) is a function such as f: R Ñ R and α P R, f(α)(x) is a function of complex variables. β Proof. Assume that α “ δ and δ ‰ 0. If f (x) ě 0, the obtained results are positive, and they constitute the real part of complex numbers. The rational/irrational order derivative of f (x) is:

1 α´1 β´δ β´δ f pxq f pxq δ f pxq δ f pxq f pαq “ “ f 1pxq “ f 1pxq xα´1 xβ´δ d x c ˆ ˙ If the rational/irrational derivative is a function of complex variables, then f (α)(x) = g(x) + ih(x), ? where i “ ´ 1. If f (x) < 0, there will be two cases: Case 1: Assume that δ is odd. Mathematics 2016, 4, 30 14 of 15

β δ β δ f pxq ´ f pxq ´ If x ě 0 or x ă 0, then the´ obtained¯ function´f (α)(¯x) is a real function and h(x) = 0 for both cases since the multiplication of any negative number in odd steps yields a negative number. Case 2: Assume that δ is even. β´δ f pxq (α) If x ě 0, then h(x) = 0 and f (x) is a real function. β δ ´ f pxq ¯ ´ If x ă 0, then the multiplication of any number in even steps yields a positive number for real´ numbers.¯ However, it yields a negative result for complex numbers, so, h(x) ‰ 0. This means that f (α)(x) is a complex function. In fact, f (α)(x) is a complex function for both cases because h(x) = 0 for some situations. Problem 1. The derivative operator D is a linear operator. The theoretical and application results presented in the previous sections demonstrated that the derivative operator does not have to be linear. The ordinary derivative operator D is linear; however, the rate of change of any function versus its independent variables does not have to be linear. This is in need of theoretical veriﬁcation. Problem 2. A new approach for derivatives was expressed in this paper. This new approach has the potential to change the integration rules in the case of orders of derivatives different from 1. The new rules for integration should be deﬁned. Problem 3. The geometrical meanings of ordinary derivatives are known. The geometrical meanings of this new approach for derivatives in the case of rational and irrational orders of derivatives are still unknown.

5. Conclusions The ordinary deﬁnition of a derivative from Newton and l’Hôpital can be considered as order = 1; in this case, the derivative operator D is linear. This paper includes a new approach for the derivative concept in which the derivative operator is not linear. The following consequences can be put forth:

(a) All functions have maximum rates of changes at inﬂection points in the positive direction. (b) All functions have minimum rates of changes at inﬂection points in the negative direction. (c) The derivative operator does not have to be linear, since the rates of changes of functions are not linear. (d) This new approach needs to prove geometrical meaning for derivative orders different from 1. (e) This new approach brought out a new problem: how to handle the integration in cases of derivative orders different from 1. (f) This new approach reveals that derivative and complex numbers have relationships.

Conﬂicts of Interest: The author declares no conﬂict of interest.

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