Tangent space to moduli space of sheaves

Let $X$ be a projective variety. Suppose that the moduli space $\mathcal M$ of nice sheaves with fixed rank and Hilbert polynomial exists. Then

$$T_{[E]}\mathcal M=\mathrm{Ext}^1(E, E).$$

For a partial proof see
Hartshorne Lectures on Deformation Theory, theorem 2.6

From MO:253578.