Let S be a semiprime ring. A function \({\Theta }:{S}\rightarrow {S}\) is defined as a multiplicative (generalized)-derivation if there is a function \({\eta }\) on S such that \({\Theta }({w}{u})={\Theta }({w}){u}+{w}{\eta }({u})\) for every \({w},{u}\in {S}\) . Assume \({\tau _1},{\tau _2},{\Gamma _1},{\Gamma _2}:{S}\rightarrow {S}\) are any functions, and for every \({w},{u}\in {S},\) \(\{{w},{u}\}^{\lambda _1}_{{\tau _1},{\tau _2}}={\tau _1}({w}){u}+{\lambda _1} {u}{\tau _2}({w}), \text { where } {\lambda _1}\in \{\pm 1\}.\) The aim of this article is to explore the following functional identities with a non zero function \({\eta }\) : (i) If \({\Gamma _2}\) is \({\tau _2}\) -commuting and \({\Theta }(\{{w},{u}\}^{\lambda _1}_{{\tau _1},{\tau _2}})={\Gamma _1}({w})\{{w},{u}\}_{{\tau _1},{\tau _2}}^{\lambda _2} {\Gamma _2}({w})\) for every \({w},{u}\in {S},\) and \({\lambda _1},{\lambda _2}\in \{\pm 1\}\) , then \({\eta }{\tau _2}\) is \({\tau _1}\) -commuting. (ii) If \({\Theta }(\{{w},{u}\}^{\lambda _1}_{{\tau _1},{\tau _2}})=\{{\Theta }({w}),{u}\}^{{\lambda _2}}_{{\tau _1},{\tau _2}}\) for every \({w},{u}\in {S}\) and \({\lambda _1},{\lambda _2}\in \{\pm 1\}\) , then \({\eta }{\tau _2}\) is \({\tau _1}\) -commuting. (iii) If \({\Theta }(\{{w},{u}\}^{\lambda _1}_{{\tau _1},{\tau _2}})=\{{w},{\Theta }({u})\}^{{\lambda _2}}_{{\tau _1},{\tau _2}}\) for every \({w},{u}\in {S}\) and \({\lambda _1},{\lambda _2}\in \{\pm 1\}\) , then \({\eta }{\tau _2}\) is \({\tau _2}\) -commuting when \({\lambda _1}={\lambda _2}\) ; otherwise, \({\eta }{\tau _2}\) is skew \({\tau _2}\) -commuting. Furthermore, we present examples to support the hypothesis of our results.
