Derivation at a point ($v:C_p^{\infty }(M)\to \mathbb{R}$)

At $p$ on a smooth manifold, a derivation at that point is a real-linear map $v:C_p^{\infty }(M)\to \mathbb{R}$ on germs of smooth functions with $v(fh)=f(p)v(h)+h(p)v(f)$. It defines a tangent vector. The local factorization $f-f(p)={\sum }_i(x^i-x^i(p))f_i$ gives $v(f)={\sum }_{i}v(x^i){\partial }_{i}f(p)$, so the coordinate derivations form a basis of the tangent space.