Is the set of all differentiable functions a vector space?

Is the set of all differentiable functions $f: \Re \rightarrow \Re$ such that $f'(0) = 0$ is a vector space over $\Re$? I was given the answer yes by someone who is better at math than me and he tried to explain it to me, but I don't understand. I am having difficulty trying to conceptualize this idea of vector spaces with functions because I can't really visualize it like a plane in 3d space. I am also wondering what is the importance of having vector spaces set over a field? It seems trivial or maybe its just me being brainwashed by years of elementary mathematics

• It's not trivial, but it does also sound like you have gotten locked into an incorrect mental model of a "vector space" by your education. Vector spaces are a lot more than just a way to describe 3D Euclidean space. – David K Feb 4 '17 at 19:21
• This may be relevant: math.stackexchange.com/questions/1847997 – David K Feb 4 '17 at 19:27