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.

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