[tex]f(x) = x+e^x \\ \\ g(x) = f^{-1}(x)\\ \\g\Big(f(x)\Big)= x \\ \\ \Big[g\Big(f(x)\Big)\Big]'= x'\\ \\\text{Aplic regula lantului pentru derivarea functiilor compuse.}\\ \\ g'\Big(f(x)\Big)\cdot f'(x) = 1 \\ \\ g'\Big(f(x)\Big)\cdot (1+e^x) = 1 \\ \\ g'(x+e^x)\cdot (1+e^x) = 1\\ \\ \text{Facem }x = 0: \\ \\ \Rightarrow g'(0+e^0) \cdot(1+e^0) = 1 \\ \\\Rightarrow g'(1)\cdot 2 = 1 \\\\ \Rightarrow \boxed{g'(1)=\dfrac{1}{2}}[/tex]
