Let $f: R \rightarrow R$ and $g: R \rightarrow R$ be two non-constant differentiable functions. If $\mathrm{f}^{\prime}(\mathrm{x})=\left(\mathrm{e}^{(\mathrm{f}(\mathrm{x})-\mathrm{g}(\mathrm{x}))}\right) \mathrm{g}^{\prime}(\mathrm{x})$ for all $\mathrm{x} \in \mathrm{R}$, and $f(1)=g(2)=1$, then which of the following statement(s) is (are) TRUE ?