On the quantum differentiation of smooth real-valued functions

<b>Abstract</b>. 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.<br> <b><br>arXiv</b>:1705.02516 <br><b>MSC 2010</b>: 26A24, 05A30, 41A58<br> <b>DOI</b>: 10.6084/m9.figshare.4956299 <br><b>Keywords</b>: derivative, differential calculus, differentiation, Taylor's theorem, Taylor's formula, Taylor's series, Taylor's polynomial, power function, Binomial theorem, smooth function, real calculus, Newton's interpolation formula, finite difference, q-derivative, Jackson derivative, q-calculus, quantum calculus, (p,q)-derivative, (p,q)-Taylor formula, mathematics, math, maths, science, arxiv, preprint, математика, наука, analysis, differential equations, quantum algebra, algebra, q-difference, algebra, analysis, calculus, open science