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Generalized Derivative: Overview, Examples

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The relatively new concept of a “generalized derivative” is an extension of the ordinary derivative to some classes of non-differentiable functions. Several generalizations of the nth derivative are obtained by one limiting process applied to one of the nth differences of f(x) [1].

The limiting process can be defined as follows [2]:

The generalized derivative of order 0 < α ≤ 1 of a real-numbered function f on a closed interval [a, b] is
generalized derivative definition

Where F(α, t) is a positive, absolutely continuous function.

If the limit described above exists, then the function is α differentiable. Different local derivatives are particular cases of the kernel F(t, α).

The notation for the generalized derivative is a bar over the derivative, like f ′(x) [3].



Types of Generalized Derivative

The collection of all generalized derivatives is the infinitesimal space. Some of the more common types of generalized derivative:

  • Peano derivatives,
  • A derivatives,
  • Symmetric (Peano) derivatives,
  • Symmetric Riemann derivatives,
  • Symmetric quantum derivatives,
  • Quantum symmetric Riemann derivatives.

There are a couple of different generalizations: Defining higher order derivatives via polynomial approximations, or as difference quotients [4].

The Peano derivative is an example of a generalized derivative that uses polynomial approximations. The first ordinary derivative and the Peano derivative have the same definition. However, the second derivatives (and higher) have different definitions. For example, the second Peano derivative is defined by [4]:
second peano derivative

The nth symmetric Riemann derivative uses difference quotients instead of polynomial approximations. For example, consider the following:
example of generalized derivative

If Δn is the nth symmetric difference, then the nth Riemann derivative is:
nth Riemann derivative

As a specific example of a generalized derivative, Dirac’s delta function is the second generalized derivative of the continuous function
generalized derivative example

References

[1] Haslam-Jones, U. (1953). On a Generalized Derivative. Retrieved August 6, 2021 from: https://academic.oup.com/qjmath/article-abstract/4/1/190/1568348?redirectedFrom=fulltext
[2] Zhao, D. and M. Luo. (2017) General conformable fractional derivative and its
physical interpretation, Calcolo, 54, 903-917
[3] Farassat, F. (1996). Introduction to Generalized Functions With Applications in Aerodynamics and Aeroacoustics. NASA Technical Paper 3428.
[4] Ash, J. Remarks on various generalized derivatives. Retrieved August 6, 2021 from: https://condor.depaul.edu/~mash/AshCalixtoGenDivSV.pdf

CITE THIS AS:
Stephanie Glen. "Generalized Derivative: Overview, Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/generalized-derivative/
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