Example
The derivative of the function (for some positive integer ) in the classical calculus is . The corresponding expressions in q-calculus and h-calculus are
with the q-bracket
and
respectively. The expression is then the q-calculus analogue of the simple power rule for positive integral powers. In this sense, the function is still nice in the q-calculus, but rather ugly in the h-calculus – the h-calculus analog of is instead the falling factorial, One may proceed further and develop, for example, equivalent notions of Taylor expansion, et cetera, and even arrive at q-calculus analogues for all of the usual functions one would want to have, such as an analogue for the sine function whose q-derivative is the appropriate analogue for the cosine.
Read more about this topic: Quantum Calculus
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“Our intellect is not the most subtle, the most powerful, the most appropriate, instrument for revealing the truth. It is life that, little by little, example by example, permits us to see that what is most important to our heart, or to our mind, is learned not by reasoning but through other agencies. Then it is that the intellect, observing their superiority, abdicates its control to them upon reasoned grounds and agrees to become their collaborator and lackey.”
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