# On the quantum differentiation of smooth real-valued functions

## TL;DR

This paper introduces a quantum-inspired approach to differentiate smooth real-valued functions by employing $q$-difference operators, Newton's interpolation, and power differences, providing new methods for analyzing derivatives within convergence radii.

## Contribution

It presents a novel quantum differentiation method using $q$-difference operators and extends Newton's interpolation to discrete analogs of Taylor series for smooth functions.

## Key findings

- Develops a $q$-difference operator for differentiation.
- Introduces $(P,q)$-power difference in Section 5.
- Shows interpolation using $q$-difference and power differences.

## Abstract

Calculating the value of $C^{k\in\{1,\infty\}}$ class of smoothness real-valued function's derivative in point of $\mathbb{R}^+$ in radius of convergence of its Taylor polynomial (or series), applying an analog of Newton's binomial theorem and $q$-difference operator. $(P,q)$-power difference introduced in section 5. Additionally, by means of Newton's interpolation formula, the discrete analog of Taylor series, interpolation using $q$-difference and $p,q$-power difference is shown.