In the definition of the Tangent space under definition via derivations
Pick a point $x$ in $M$. A derivation at $x$ is a linear map $D : C^{\infty}(M) \to \mathbb{R} $ which has the property that for all $ƒ, g$ in $C^{\infty}(M)$: $D(fg) = D(f)\cdot g(x) + f(x)\cdot D(g)$ modeled on the product rule of calculus. These derivations form a real vector space in a natural manner; this is the tangent space $T_xM$
My question is, it is given that $D : C^{\infty}(M) \to \mathbb{R}$ is real valued and gives out scalar and how does $D(fg)$ forms a vector space ?
Same is the case in the next sentence as well.