Shorter Notes: A Variant of the Chain Rule for Differential Calculus

1981 ◽  
Vol 81 (1) ◽  
pp. 155
Author(s):  
Richard A. Graff
Keyword(s):  
Differential Calculus ◽  
Chain Rule

scholarly journals A Framework for Differential Calculus on Persistence Barcodes

2021 ◽  
Author(s):  
Jacob Leygonie ◽  
Steve Oudot ◽  
Ulrike Tillmann
Keyword(s):  
Data Analysis ◽  
Differential Calculus ◽  
Gradient Descent ◽  
Chain Rule ◽  
Topological Data Analysis ◽  
Objective Functions ◽  
A Chain ◽  
Topological Data

AbstractWe define notions of differentiability for maps from and to the space of persistence barcodes. Inspired by the theory of diffeological spaces, the proposed framework uses lifts to the space of ordered barcodes, from which derivatives can be computed. The two derived notions of differentiability (respectively, from and to the space of barcodes) combine together naturally to produce a chain rule that enables the use of gradient descent for objective functions factoring through the space of barcodes. We illustrate the versatility of this framework by showing how it can be used to analyze the smoothness of various parametrized families of filtrations arising in topological data analysis.

Providing Motivation for the Chain Rule

1967 ◽  
Vol 60 (8) ◽  
pp. 850-855
Author(s):  
Lyman S. Holden
Keyword(s):  
General Formula ◽  
Differential Calculus ◽  
Geometric Interpretation ◽  
Chain Rule

The purpose of this article is to describe how a certain graphing technique can be used to increase the heuristic appeal of a general formula from the differential calculus. The formula to which reference is made is the formula for the derivative of the composition of two functions (the chain rule). We shall interpret this rule in relation to the graph of the function that is the composition of two functions. Using the geometric interpretation of the derivative as the slope of a curve, it will be possible to provide improved motivation for the rule.

scholarly journals Second-order neutrosophic boundary-value problem

2021 ◽  
Author(s):  
Sandip Moi ◽  
Suvankar Biswas ◽  
Smita Pal(Sarkar)
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Differential Calculus ◽  
Boundary Value ◽  
Second Order ◽  
Numerical Examples ◽  
Different Types ◽  
First Time

AbstractIn this article, some properties of neutrosophic derivative and neutrosophic numbers have been presented. This properties have been used to develop the neutrosophic differential calculus. By considering different types of first- and second-order derivatives, different kind of systems of derivatives have been developed. This is the first time where a second-order neutrosophic boundary-value problem has been introduced with different types of first- and second-order derivatives. Some numerical examples have been examined to explain different systems of neutrosophic differential equation.

On students’ understanding of the differential calculus of functions of two variables

2015 ◽  
Vol 38 ◽  
pp. 57-86 ◽  
Author(s):  
Rafael Martínez-Planell ◽  
Maria Trigueros Gaisman ◽  
Daniel McGee
Keyword(s):  
Differential Calculus ◽  
Functions Of Two Variables
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