✔ A Simple Derivative For Computer Applications! (2010)

This theory's related publications: http://amir-shachar.com/gpage.html

In Calculus and discrete mathematics, the 'Detachment' operator (originated in 2010) is a discrete type of a function's Derivative. Other than forming a numerical optimization to the derivative (as discussed in the video), this operator and its extensions also hold theoretical (see the publications) and pedagogical advantages.

In simple word, the Detachment operator examines a function's trend of change - whether it is ascending, descending or constant (while the Derivative inquires more information, and examines the function's rate of change). One might think that the Detachment is a "special case" of the Derivative, in the sense that once we know the function's rate of change, we immediately know also its trend of change. However, that intuition is not always correct, and this video depicts some of the Detachment's advantages, namely:

- The Detachment operator is Defined for non differentiable and discontinuous functions.
- It supplies information on the function's monotony behavior in cases where the derivative vanishes (either zeroed or non-existent).
- This operator, and its extensions, are numerically efficient (in terms of the evaluation of the limit via a computer according to the definition of the limit).
- It is defined in advanced theories of classical analysis, where the derivative is undefined (such as metric spaces).
- It might help students to better grasp the idea of the Derivative in Calculus.
- It combines between two types of mathematics: continuous math (Calculus) and discrete math (since the sign function accepts only a finite number of values).

You might also want to check out the demonstration (full disclosure: uploaded by me), at Wolfram's Demonstrations Project on the geometric interpretation of the basic definition:
http://www.youtube.com/watch?v=WRAB9CLUbww

Feel free to contact me to discuss those ideas, or any other mathematical thoughts you may have: amir.shachar@mail.huji.ac.il

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