Using knowledge of derivatives, what is $f(x)$ if $f'(x) = 4x + e^x$ with initial condition $f(0)=2$? How to solve this with knowledge of derivatives? Is it asking to solve this without using anti-derivatives?

Probably this is the intention of the exercise: because $f'(x)=g(x)+h(x)$ then $f(x)=G(x)+H(x)+C$ for some constant $C$ where $G$ and $H$ are antiderivatives of $g$ and $h$ respectively.
I mean: "knowledge of derivatives" seems to me to search for such $G$ and $H$ with derivatives $g$ and $h$, knowing that if $f(x)=G(x)+H(x)+C$ then $f'(x)=g(x)+h(x)$.
Then for any valid pair $G$ and $H$ the constant $C$ is defined by the value $f(0)=2$.
