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The Properties of Derivatives
Published in James K. Peterson, Basic Analysis I, 2020
(f/g)′=(f×1/g)′=f′×1/g+f×(−g′/g2)by the product rule. Getting a common denominator, we have(f/g)′=f′g−fg′g2.
Treatment of Geometric Nonlinearities
Published in Michael R. Gosz, Finite Element Method, 2017
It turns out that the change operator, D, obeys the usual properties of differentiation. Thus, for example, we can use the product rule of differentiation in order to compute the change of products of functions, i.e., D(Aℬ)=(DA)(ℬ)+(A)(Dℬ)
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).
Generalized derivatives of computer programs
Published in Optimization Methods and Software, 2022
Matthew R. Billingsley, Paul I. Barton
As shown by Khan and Barton [17], given open sets and , and functions and that are L-smooth at and , respectively, the composition is L-smooth at . Furthermore, for any , the following chain rule for LD-derivatives is satisfied: The product rule is a special case of the chain rule (1) (with f a scalar function):
Solutions of local fractional sine-Gordon equations
Published in Waves in Random and Complex Media, 2019
H. Karayer, D. Demirhan, F. Buyukkilic
Many rational systems in physics, engineering and other sciences can be described successfully using fractional order derivatives and integrals [6]. Applications of this theory have been used in modeling control systems [7], heat transfer [8], fluid dynamics [9]. However, the theory contains additional difficulties due to the definitions of fractional order derivative and integral operators. The prevalently well known of these operators are given as Riemann–Liouville, Caputo, Riesz fractional derivative operators which do not provide the basic rules in standard calculus like product rule, quotient rule, chain rule [9]. For this reason some numerical techniques are required in order to solve fractional order differential equations. Solutions of nonlinear fractional partial differential equations can be obtained by using homotopy analysis method (HAM) which is a semi-analytic method [10]. Radial basis functions (RBF) method can be used in order to achieve numerical solutions of fractional order partial differential equations [11]. Recently, a new fractional order derivative operator namely conformable fractional derivative operator which allows us to obtain analytical solutions of fractional order differential equations, is defined by Khalil et al. [12];