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Differential Calculus is a staple of the college mathematics major's diet. Eventually one becomes tired of the same routine, and wishes for a more diverse meal. The college math major may seek to generalize applications of the derivative…

For function of one variable, differentiability is equivalent to the existence of tangent line as the limit of secant line. The genuine counterpart of this equivalence for function of several variables is obtained for the first time.

Motivated by extending the functional stochastic calculus, to important functionals to which it does not apply, a notion of functional derivative along a curve is introduced. This new setting is developed by incorporating path-dependent…

Derivative polynomials in two variables are defined by repeated differentiation of the tangent and secant functions. We establish the connections between the coefficients of these derivative polynomials and the numbers of interior and left…

A function in a class $\mathcal{F}(X)$ is said to be subdifferentially determined in $\mathcal{F}(X)$ if it is equal up to an additive constant to any function in $\mathcal{F}(X)$ with the same subdifferential. A function is said to be…

Diagrams as a graphic expresion of derivatives is proposed for calculation of derivatives for composed function. The concret diagram is understood as a virtual derivative in contrast of concret derivative. In polynomial expression of…

The fractional calculus is useful to model non-local phenomena. We construct a method to evaluate the fractional Caputo derivative by means of a simple explicit quadratic segmentary interpolation. This method yields to numerical resolution…

In this paper we introduce the concept of \emph{multivector functionals.} We study some possible kinds of derivative operators that can act in interesting ways on these objects such as, e.g., the $A$-directional derivative and the…

We consider a scalar-valued implicit function of many variables, and provide two closed formulae for all of its partial derivatives. One formula is based on products of partial derivatives of the defining function, the other one involves…

Ambiguity is shown in the context of the differential calculus of several variables and with the help of the language of category theory, a way to solve it in its most general form is offered. It is also shown that this new definition is…

A method is proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra. As an example the generalized quantum plane is studied. It is found that there is a strong correlation, but not a…

Universal geometric calculus simplifies and unifies the structure and notation of mathematics for all of science and engineering, and for technological applications. This paper treats the fundamentals of the multivector differential…

This article provides an accessible introduction to fractional derivatives, a concept that extends classical calculus by allowing derivatives of non-integer order. It explores both the fundamental definitions and some of the most relevant…

The discrete, the quantum, and the continuous calculus of variations, have been recently unified and extended by using the theory of time scales. Such unification and extension is, however, not unique, and two approaches are followed in the…

In this paper the analogy between differential forms arising from integrals in additive calculus and forms arising from the integrals in product calculus is investigated. It is found that with an appropriate definition of scalar…

This short article contains the construction of a construction that generalizes the concept of the derivative of a function of one variable, using the theory of filters. The paper presents a new concept, demonstrates that it really…

We pose a new algebraic formalism for studying differential calculus in vector bundles. This is achieved by studying various functors of differential calculus over arbitrary graded commutative algebras (DCGCA) and applying this language to…

Consider a real valued function defined, but not differentiable at some point. We use sequences approaching the point of interest to define and study sequential concepts of secant and cord derivatives of the function at the point of…

We discuss a version of the fundamental theorem of calculus in several variables and some applications, of potential interest as a teaching material in undergraduate courses.

This paper is an introduction to the theory of multivector functions of a real variable. The notions of limit, continuity and derivative for these objects are given. The theory of multivector functions of a real variable, even being similar…