Quotient Rule Research Articles

Misconceptions and Errors Among Grade 12 Students When Learning Differentiation Rules: A Case Study

t is an established fact that some grade 12 students learning differentiation in calculus are found struggling with the rule of differentiation and these rules include power rule, quotient rule, chain rule and product rule. Because of these, some students came up with some misconceptions which eventually resulted to the students having multiple errors when learning rules of differentiation. The reason associated with these common errors are not far from their failure to model some prerequisite knowledge in the laws of logarithm into their learning of the laws of differentiation. And this has resulted to the poor performance of some students in mathematics (calculus) since calculus carries about 40% in the overall grade in mathematics as a subject at grade 12 level. On this note, this paper presents an analysis of students’ errors and misconceptions in learning differentiation rule. The study was conducted among grade 12 students preparing for NSC examination in a high school in Limpopo province in South Africa. And the data was initially collected using 35 test scripts of grade 12 on the topic of differentiation and differentiation rule. A qualitative approach was considered, and the data collected was analysed, focusing on product rules, quotient rules and chain rules and the errors committed. The result of the study indicated that some grade 12 students make some errors and misconceptions in differentiation rules. And these were because of poor conceptual understanding, poor mathematics language understanding, and some other error. It was confirmed that the results of this study highlight the common mistakes and errors students make when learning differentiation rules, and these errors are; conceptual, systemic, language, and generalization errors.

A new fractional derivative extending classical concepts: Theory and applications

In this paper, a novel general definition for the fractional derivative and fractional integral based on an undefined kernel function is introduced. For 0<α≤1, this definition aligns with classical interpretations and is applicable for calculating the derivative in an open negative interval I⊆[a,+∞),a∈R. Additionally, when α=1, the definition coincides with the classical derivative. Fundamental properties of the fractional integral and derivative, including the product rule, quotient rule, chain rule, Rolle’s theorem, and the mean value theorem, are derived. These properties are illustrated through various applications to demonstrate their applicability. Furthermore, some applications of solving fractional nonlinear systems of integro-differential equations using framelets are presented.

A general form of Gronwall inequality with Stieltjes integrals

We present a new Gronwall inequality for Stieltjes integrals, which improves numerous existing results, and has a simple proof based on the quotient rule for Stieltjes integrals. As an application, we obtain uniqueness theorems for measure differential equations and nabla dynamic equations. Finally, we revisit the topic from the perspective of Stieltjes derivatives.

Why Isn't There an Inverse Quotient Rule?

When learning Calculus, students are taught how to undo the Chain Rule through u-Substitution and may have been introduced to Integration by Parts as a way to undo the Product Rule. This paper describes analogous integration techniques to undo the Quotient Rule and argues that the “Inverse Quotient Rule” is a worthwhile technique to teach in Integral Calculus. An appendix includes 26 textbook-style exercises for interested students.

Improved Network-Calculus Nodal Delay-Bounds in Time-Sensitive Networks

In time-sensitive networks, bounds on worst-case delays are typically obtained by using network calculus and assuming that flows are constrained by bit-level arrival curves. However, in IEEE TSN or IETF DetNet, source flows are constrained on the number of packets rather than bits. A common approach to obtain a delay bound is to derive a bit-level arrival curve from a packet-level arrival curve. However, such a method is not tight: we show that better bounds can be obtained by directly exploiting the arrival curves expressed at the packet level. Our analysis method also obtains better bounds when flows are constrained with g-regulation, such as the recently proposed Length-Rate Quotient rule. It can also be used to generalize some recently proposed network-calculus delay-bounds for a service curve element with known transmission rate.

Results on Katugampola Fractional Derivatives and Integrals

In this paper, we introduce and develop a new definitions for Katugampola derivative and Katugampola integral. In particular, we defined a (left) fractional derivative starting from a of a function f of order α∈(m-1, m] and a (right) fractional derivative terminating at b, where m ∈ N. Then, we give some proprieties in relation to these operators such as linearity, product rule, quotient rule, power rule, chain rule, and vanishing derivatives for constant functions.

A note on the derivation of quotient rules and their use in QR kinematics

A note on the derivation of quotient rules and their use in QR kinematics

Gohar Fractional Derivative: Theory and Applications

The local fractional derivatives marked the beginning of a new era in fractional calculus. Due to their that have never been observed before in the field, they are able to fill in the gaps left by the nonlocal fractional derivatives and substantially increase the field’s theoretical and applied potential. In this article, we introduce a new local fractional derivative that possesses some classical properties of the integer-order calculus, such as the product rule, the quotient rule, the linearity, and the chain rule. It meets the fractional extensions of Rolle’s theorem and the mean value theorem and has more properties beyond those of previously defined local fractional derivatives. We reveal its geometric interpretation and physical meaning. We prove that a function can be differentiable in its sense without being classically differentiable. Moreover, we apply it to solve the Riccati fractional differential equations to demonstrate that it provides more accurate results with less error in comparison with the previously defined local fractional derivatives when applied to solve fractional differential equations. The numerical results obtained in this work by our local fractional derivative are shown to be in excellent agreement with those produced by other analytical and numerical methods such as the enhanced homotopy perturbation method (EHPM), the improved Adams-Bashforth-Moulton method(IABMM), the modified homotopy perturbation method (MHPM), the Bernstein polynomial method (BPM), the fractional Taylor basis method (FTBM), and the reproducing kernel method (RKM).

The Integration by Quotient Rule

The Integration by Quotient Rule

A New conformable fractional derivative and applications

The motivation for this paper comes from other papers treating the fractional derivatives. We introducea new definition of fractional derivative which obeys classical properties including linearity, product rule,&nbsp;quotient rule, power rule, chain rule, Rolle’s theorem, mean value theorem and Taylor series. Usage of thedefined derivative is given in the example section which shows how our derivative can be used in solving&nbsp;differential equations. Comparison of our derivative with the derivative defined by Abdejjawad and overall&nbsp;conclusions are given in the conclusion section.

Exploring Errors and Misconceptions in Differentiation: A Case Study of Advanced Level Students in Zimbabwe

Students’ errors help teachers to tease out misconceptions and decide on what intervention strategies to adopt and use to challenge these alternative conceptions. The purpose of the study was to explore A-Level students’ errors and misconceptions when solving problems in differentiation. A largely qualitative case study strategy was adopted. The case was made up of Two A-Level teachers and 25 students. After covering the topic differentiation, a test was administered. Qualitative data was collected using the test scripts and syllabus documents, questionnaire and focus group discussions. Content analysis was used to reveal errors and misconceptions when solving problems in differentiation. The common errors and misconceptions displayed by A-Level students were largely procedural when they failed to use the quotient rule, chain rule and power rule in finding derivatives. Our study found out that errors and misconceptions when using the power rule were adding a one to the power part instead of subtracting, bringing down the power and mistakenly adding a one to the power, failure to integrate functions, failure to separate variables correctly, failure to find the required derivative and using wrong laws to solve problems in differentiation. These errors and misconceptions were, possibly originating from students’ lack of prior knowledge of differentiation, laws of logarithms, proportionality and integration and rate of change. We recommend further research using larger population and samples.

A Study on Internship Student Debriefing Materials in ‘Merdeka Belajar Kampus Merdeka’ Program

&lt;em&gt;The VUCAS era as it is today has brought about changes in attitudes and human interactions at work. A person’s competancy is demanded to be acquired more than one or what is so called, multitasking. Whilst, higher education Institutions does not prepare these competencies for graduate students. The industrial world is also overwhelmed by the quality of untrained new employees. There for, it is esential to create program in order to link and match between Higher Education Institutions Programs and the need of industrial world. The Ministry of Education and Culture has created a 3 semesters training program outside its study program, one of which is an internship in companies. Due to this, we design debriefing material in accordance to the industrial perspective. This research aims to obtain perspectives on the material from the industry. This qualitative research was conducted using in-depth interviews, both in person and online to gather information from industry players. The results obtained will be materials that must be given to students during the debriefing, there are two important materials to be given, namely; knowledge and work attitude. Knowledge includes: company organization, job duties and responsibilities, remuneration systems, industrial relations, employee development and management systems. Work attitudes include general values such as integrity, discipline, leadership, team work, interpersonal skills, analitical competency, open mindedness, emotional quotient (EQ), burning desire/passion and time management skills and rules in work (manner).&lt;/em&gt;

A New Conformable Fractional Derivative and Applications

This paper is motivated by some papers treating the fractional derivatives. We introduce a new definition of fractional derivative which obeys classical properties including linearity, product rule, quotient rule, power rule, chain rule, Rolle’s theorem, and the mean value theorem. The definition D α f t = lim h ⟶ 0 f t + h e α − 1 t − f t / h , for all t &gt; 0 , and α ∈ 0,1 . If α = 0 , this definition coincides to the classical definition of the first order of the function f .

Calculation of vectorial derivatives for functions of a quaternion variable and their properties

Various attempts have been made in defining the derivative of a quaternionic function due to the noncommutativity of the product over quaternions. We observe that the difference in the left and right operations caused by the noncommutativity of the quaternion product is determined by the vector part of the quaternion. In this paper, we propose a corresponding derivative to replace the derivative of a quaternion-valued function of a quaternionic variable using the component terms of a quaternion. Further, the analogous constant, product, and quotient rules for the proposed calculations are given. Application of the proposed derivatives is provided to compute the derivatives of elementary functions. Several illustrations are also presented.

Study of Fractional Analytic Functions and Local Fractional Calculus

In this present paper, the role of fractional analytic function in local fractional calculus is studied. Some important properties and theorems in local fractional calculus are discussed, such as product rule, quotient rule, chain rule, fundamental theorem of local fractional calculus, change of variable, integration by parts and so on. In addition, we propose several examples and formulas of local fractional calculus.

Prioritization of Students Understanding in Rules of Differentiation using Fuzzy Soft Matrices and Lambda – max Method

Calculus is one of the most important courses especially for undergraduate students in many fields of study. Some researchers have identified the causes of the high failure rate which includes lack of basic foundation of mathematics and basic concept of differentiation. Aside from that, the main problems that can be seen among students are the difficulty in identifying the type of function in differentiation and identifying the suitable method to solve a particular problem. There are three rules included in differentiation which are the Chain Rule, Product Rule and Quotient Rule. This study is conducted to examine the level of understanding of the students on the function and the derivative techniques after applying derivative game applications in this course. This paper is based on Fuzzy Analytic Hierarchy Process (Fuzzy AHP) which use fuzzy number in pair-wise comparison matrix. The prioritization of students’ understanding in differentiation rules will then be measured by Fuzzy AHP using Lambda-max method. The highest among the three rules in differentiation will considered as a result. The findings of this study indicated that the highest score with 0.4700 is the Chain Rule. This study can help lectures to know the level or understanding among three rules in differentiation and lecturer well prepared their teaching materials in the classroom as well as to reduce the failure rate among students in this course

Numerical Simulation of Non-cooperative and Cooperative Equilibrium Solutions for a Stochastic Government Debt Stabilization Game

In this article, we consider the impact uncertainty has on policies and realization of targets aimed at the stabilization of government debt. The problem is motivated by the fact that in many countries revenues are to a large extent uncertain. Following Tabellini (TEDC 4: 427–442, 1986), we model the debt stabilization problem as a dynamic game between government and central bank. However, different from Tabellini (TEDC 4: 427–442, 1986) we assume that nominal income in this problem setting is uncertain and model its price dynamics by a stochastic differential equation. By employing the quotient rule from stochastic calculus, government debt, primary fiscal deficit, and base money are scaled by this nominal income process. Then, the debt stabilization problem is formulated within this stochastic framework. Assuming a feedback information structure we solve the problem both within a non-cooperative and cooperative mode of play. To explore the effects of uncertainty on government and central bank polices and debt stability, we perform a simulation study and sensitivity analysis. One of the major conclusions from this study seems to be that cooperation has a smoothing effect on pursued policies but that this does not necessarily lead on average to a better tracking of the debt target. Furthermore, we observe that the more uncertainty there is, the more policymakers are inclined to emphasize the stabilization of debt.

A new generalized θ-conformable calculus and its applications in mathematical physics

In this work, a new generalized concept of conformable derivative is given and named generalized θ–conformable derivative (GTCD). Several properties are studied such as the chain rule, the quotient rule, the product rule, the Rolle’s theorem, the mean value theorem and the fundamental theorems of calculus. Also, the geometrical and physical interpretations of the GTCD are presented. It is very easy to see that the GTCD is comprehensive and includes many past derivatives as special cases. An application of the generalized θ–conformable derivatives (GTCDs) is presented along with the –approach to construct a novel gathering of exact solutions for the nonlinear (2+1)-dimensional biological population model (BPM) involving GTCDs. Moreover, some comparisons and graphical interpretations are given to support our results related to the (2+1)-dimensional BPM. All acquired results regarding the GTCD show its validity to applying in many problems in mathematical physics, engineering, biology and others.

The mixed conformable partial derivatives

In this paper we introduce a new conformable derivative call it mixed conformable partial derivative, which obeys classical properties, including linearity, product rule, quotient rule and vanishing derivatives for constant functions. As an application, we establish an Opial type inequality for the mixed second order conformable partial derivatives.

Calculus Identities for Generalized Simplex Gradients: Rules and Applications

Simplex gradients, essentially the gradient of a linear approximation, are a popular tool in derivative-free optimization (DFO). In 2015, a product rule, a quotient rule, and a sum rule for simplex gradients were introduced by Regis [Optim. Lett., 9 (2015), pp. 845--865]. Unfortunately, those calculus rules only work under a restrictive set of assumptions. The purpose of this paper is to provide new calculus rules that work in a wider setting. The rules place minimal assumptions on the functions involved and the interpolation sets. The rules further lead to an alternative approach to gradient approximation in situations where the rules could be applied. We analyze the new approach, provide error bounds, and include some preliminary testing on numerical stability and accuracy.