Derivations and linear functions along rational functions

The main purpose of this paper is to give characterization theorems on derivations as well as on linear functions. Among others the following problem will be investigated: Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n \in \mathbb{Z}, f, g\colon\mathbb{R} \to\mathbb{R}}$$\end{document} be additive functions, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\left(\begin{array}{cc} a&b\\ c&d \end{array} \right) \in \mathbf{GL}_{2}(\mathbb{Q})}$$\end{document} be arbitrarily fixed, and let us assume that the mapping \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \phi(x)=g\left(\frac{ax^{n}+b}{cx^{n}+d}\right)-\frac{x^{n-1}f(x)}{(cx^{n}+d)^{2}} \quad \left(x\in\mathbb{R}, cx^{n}+d\neq 0\right)$$\end{document}satisfies some regularity on its domain (e.g. (locally) boundedness, continuity, measurability). Is it true that in this case the above functions can be represented as a sum of a derivation and a linear function? Analogous statements ensuring linearity will also be presented.