Let $M$ be a $k$-submanifold of $\mathbb{R}^n$ defined by the equation: $$\mathbf{F}(x_1,...,x_n)=\mathbf{0}$$
I know that a vector field on $\mathbb{R}^n$ restricts on a vector field on the submanifold iff it annihilates the components of $\mathbf{F}$. Basically there is a surjection
$$\Pi : \text{Ann}_{\text{Der}(C^{\infty}(\mathbb{R}^n))}(F_1,...,F_{n-k})\to \text{Der}(C^{\infty}(M))$$
that simply restricts the vector field to the submanifold. I understand what "restricting" to the submanifold means if we see vector field in the intuitive manner, but I fail to understand how a derivation of the algebra $C^{\infty}(\mathbb{R}^n)$ restricts to a derivation of the algebra $C^{\infty}(M)$. Who is $\Pi(X)$ as a derivation?