Tangent space (Derivation definition)

Let $M$ be a smooth manifold, and let $p\in M$ . A linear map $v: C^\infty(M)\to\mathbb{R}$ is called a derivation at $p$ if it satisfies $\displaystyle v(fg)=f(p)vg+g(p)vf\qquad\text{for all}\ f,g\in C^\infty(M).$
The tangent space to $M$ at $p$ , denoted by $T_pM$ , is defined as the set of all derivations of $C^\infty(M)$ at $p$ .