Generalized orthogonality equations in normed spaces

Let X, Y be real normed spaces and let $$\rho _{+}^{'},\rho _{-}^{'}$$ be norm derivatives. In this work, we solve a system of functional equations $$\begin{aligned} {\left\{ \begin{array}{ll} \rho _{+}^{'}(f(x),f(y))=g(x)\rho _{+}^{'}(x,y)\\ \rho _{-}^{'}(f(x),f(y))=g(x)\rho _{-}^{'}(x,y) \end{array}\right. } \end{aligned}$$ where the functions $$f:X\rightarrow Y$$ and $$g:X\rightarrow \mathbb {R}$$ are unknown.