{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,15]],"date-time":"2026-03-15T15:22:37Z","timestamp":1773588157710,"version":"3.50.1"},"reference-count":30,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,8,24]],"date-time":"2022-08-24T00:00:00Z","timestamp":1661299200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"FCT (Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia)","award":["UIDB\/00013\/2020"],"award-info":[{"award-number":["UIDB\/00013\/2020"]}]},{"name":"FCT (Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia)","award":["UIDP\/00013\/2020"],"award-info":[{"award-number":["UIDP\/00013\/2020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we define the \u03b2-partial derivative as well as the \u03b2-directional derivative of a multi-variable function based on the \u03b2-difference operator, D\u03b2, which is defined by D\u03b2f(t)=f(\u03b2(t))\u2212f(t)\/\u03b2(t)\u2212t, where \u03b2 is a strictly increasing continuous function. Some properties are proved. Furthermore, the \u03b2-gradient vector and the \u03b2-gradient directional derivative of a multi-variable function are introduced. Finally, we deduce the Hahn-partial and the Hahn-directional derivatives associated with the Hahn difference operator.<\/jats:p>","DOI":"10.3390\/sym14091766","type":"journal-article","created":{"date-parts":[[2022,8,25]],"date-time":"2022-08-25T01:50:26Z","timestamp":1661392226000},"page":"1766","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["The Directional Derivative in General Quantum Calculus"],"prefix":"10.3390","volume":"14","author":[{"given":"Avin O.","family":"Karim","sequence":"first","affiliation":[{"name":"Department of Mathematical sciences, College of Basic Education, University of Sulaimani, Sulaimani P.O. Box 46, Kurdistan, Iraq"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7210-9118","authenticated-orcid":false,"given":"Enas M.","family":"Shehata","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Faculty of Science, Menoufia University, Shibin El-Kom 32511, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7418-3634","authenticated-orcid":false,"given":"Jos\u00e9 Luis","family":"Cardoso","sequence":"additional","affiliation":[{"name":"Escola de Ci\u00eancias e Tecnologia, University of Tr\u00e1s-os-Montes e Alto Douro, 5000-801 Vila Real, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s13662-014-0348-8","article-title":"New concepts of fractional quantum calculus and applications to impulsive fractional q-difference equations","volume":"2015","author":"Tariboon","year":"2015","journal-title":"Adv. Differ. 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