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Introduction
Published in David V. Kalbaugh, Differential Equations for Engineers, 2017
From these two rules one can derive the quotient rule. Let y(t)=f(t)/g(t). Then from the product rule we have dydt=dfdt1g+fddt(1g)
Errors in calculus
Published in Breach Mark, Essential Maths for Engineering and Construction, 2017
A quotient is where one expression is divided by another. When differentiating a quotient, the quotient rule is used. This rule says that if y=uv and u and v are both functions of x, then dydx=vdudx-udνdxv2.
A quantitative and covariational reasoning investigation of students’ interpretations of partial derivatives in different contexts
Published in International Journal of Mathematical Education in Science and Technology, 2023
This qualitative study used task-based interviews (Goldin, 2000) to examine calculus students’ quantitative and covariational reasoning when interpreting partial derivatives in mathematics and real-world contexts. The interviews covered the five tasks presented in the theoretical perspective section. The purpose of Task 1 was to examine students’ ability to calculate first order partial derivatives using rules of differentiation such as the quotient rule and chain rule when given an equation for a real-valued function of two variables. Task 2 was designed with a three-fold purpose, namely (a) to examine students’ ability to interpret the output of a real-valued function of two variables in a mathematics context, (b) to examine students’ ability to calculate first order partial derivatives using the power rule of differentiation when given an equation for a real-valued function of two variables, and (c) to examine students’ ability to interpret first order partial derivatives in a mathematics context. Tasks 3 through 5 (adapted from Hughes-Hallett et al., 2014) were designed with a two-fold purpose, namely (a) to examine students’ ability to interpret output values of real-valued functions of two or three variables in different real-world contexts, and (b) to examine students’ ability to interpret first order partial derivatives in different real-world contexts.
Improving mathematics diagnostic tests using item analysis
Published in International Journal of Mathematical Education in Science and Technology, 2023
For the ETH s21t, we used the IRT results to identify a total of five items to remove or replace. First, we considered items that had very low discrimination (and hence information). We removed z2, z12, z13 and z25 for this reason; however, we decided to retain z1 (which stood out for having both low discrimination and difficulty) as we had always intended this to be a straightforward item to ease students into the test. Second, we decided to remove z19, as the IRT analysis had identified that its performance was very similar to z18; indeed, they were one of only two pairs of items showing signs of local dependence (the other pair being z34 and z35, which we discuss below). Both items had a similar setup, with one on the product rule (z18) and the other on the quotient rule (z19).
A Novel Conformable Fractional-Order Terminal Sliding Mode Controller for a Class of Uncertain Nonlinear Systems
Published in IETE Journal of Research, 2023
Although the FO calculus has several benefits, it has some problems too, e.g. complexity in calculations. Also, it does not obey some properties of integer operators, such as Leibniz ‘s rule, chain rule, product, and the quotient rule. So, in 2014, a conformable derivative as a simple FO derivative was introduced in [16] which did not suffer from the disadvantages of the other FO definitions. In [17], some concepts of conformable fractional-order (CFO) derivative were developed. The CFO derivative has provoked the interests of researchers [18–20]. In [21], the stability of CFO systems was investigated. In [18], the dynamic behaviours of CFO Lotka–Volterra predator–prey systems were studied. The numerical solutions of the CFO equations were presented in [19]. In addition, a CFO simplified Lorenz system is studied. CFO chaotic systems were investigated in [20].