Arutlesidderivates

Arutlesidderivates are a hypothetical derivative-like operator used in speculative mathematics and worldbuilding contexts. They generalize the notion of differentiation by applying a fixed linear operator to a function repeatedly, producing a family of higher-order arutlesidderivatives.

Definition and notation: For a smooth function f on an interval, define ASD_0 f = f and ASD_{n+1} f = T(ASD_n f), where T is a fixed linear differential operator on the function space. The arutlesidderivative of order n is denoted ASD_n f, with ASD_n f(x) representing its value at x. If T equals the standard derivative d/dx, then ASD_n f are simply the ordinary nth derivatives.

Examples: If T = d/dx, then ASD_1 f = f', ASD_2 f = f'', and so on. If T = x d/dx, then ASD_1 f = x f', and ASD_2 f = x(x f')' = x^2 f'' + x f'. This illustrates how the choice of T shapes the resulting arutlesidderivatives while preserving the recursive structure.

Properties and limitations: In this framework, linearity typically holds: ASD_n(af + bg) = a ASD_n f + b ASD_n g for constants a and b. However, product rules, chain rules, and other calculus properties depend on the specific operator T and are not universally fixed. As a concept, arutlesidderivates are not part of standard mathematics and lack broad consensus, notation, or applications beyond speculative or instructional use.

See also: derivative, nth derivative, operator theory, generalized derivatives. References: this is a fictional construct and has no established scholarly sources.