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Errors in calculus
Published in Breach Mark, Essential Maths for Engineering and Construction, 2017
When differentiating a function of a function we use the chain rule. The chain rule, at its simplest, says that if y = f (u) and u = g (x) then dydx=dydu×dudx..
Logarithmic differentiation
Published in John Bird, Bird's Higher Engineering Mathematics, 2021
Logarithmic differentiation is a means of differentiating algebraically complicated functions or functions for which the ordinary rules of differentiation do not apply. The technique is performed in cases where it is easier to differentiate the logarithm of a function rather than the function itself. Logarithmic differentiation relies on the function of a function rule (i.e. chain rule) as well as properties of logarithms (in particular, the natural logarithm, or logarithm to the base e) to transform products into sums and divisions into subtractions, and can also be applied to functions raised to the power of variables of functions. Logarithmic differentiation occurs often enough in engineering calculations to make it an important technique.
Logarithmic differentiation
Published in John Bird, Bird's Engineering Mathematics, 2021
Logarithmic differentiation is a means of differentiating algebraically complicated functions or functions for which the ordinary rules of differentiation do not apply. The technique is performed in cases where it is easier to differentiate the logarithm of a function rather than the function itself. Logarithmic differentiation relies on the function of a function rule (i.e. chain rule) as well as properties of logarithms (in particular, the natural logarithm or logarithm to the base e) to transform products into sums and divisions into subtractions, and can also be applied to functions raised to the power of variables of functions. Logarithmic differentiation occurs often enough in engineering calculations to make it an important technique.
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):
The impact of procedural and conceptual teaching on students' mathematical performance over time
Published in International Journal of Mathematical Education in Science and Technology, 2021
Vahid Borji, Farzad Radmehr, Vicenç Font
The findings also showed that students had different performances when differentiating different types of implicit functions. The main difficulty was when a term was a composite function where the inner function consisted of both and (e.g. ). Therefore, lecturers could focus more on these types of terms during teaching implicit differentiation. They could explain to students that they can decompose the function and then use the chain rule to differentiate it. For example, is the composition of the outer function and the inner function . The derivatives of and are and respectively. Using the chain rule the derivative of is .
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];