Hartshorne problem III.4.10

Question

I am stuck on this problem from Hartshorne and to be honest I don't really know where to start. The problem is the following: $X$ is a non-singular variety over an algebraically closed field $k$ and $\mathcal{F}$ is a coherent sheaf on $X$. Then we have to show that there is a one to one correspondence $$\{\text{Infinitesimal extensions of $X$ by $\mathcal{F}$}\}/\text{iso}\leftrightarrow H^1(X,\mathcal{F}\otimes \mathcal{T})$$ where $\mathcal{T}=\mathscr{H}om(\Omega_{X/k},\mathcal{O}_X)$ is the tangent sheaf. I had the following idea: Let $(X',\mathcal{J})$ be an infinitesimal extension (we may assume $X'=X$ as topological spaces). Cover $X$ by a finite number of open affine $U_i$. Then $(U_i',\mathcal{J}|_{U_i'})$ is an extension of $U_i$ by $\mathcal{F}|_{U_i}$. By problem $II.8.7$ in Hartshorne we know these extensions are trivial. So we get isomorphisms $$\varphi_i:\mathcal{O}_{U_i'}\cong \mathcal{O}_{U_i}\oplus \mathcal{F}|_{U_i}.$$ Somehow I wanted to use these isomorphisms to construct a cycle in $\breve{C}^1(\mathfrak{U},\mathcal{F}\otimes \mathcal{T})$ but I don't know how. As I said I don't know if this is the way to start, any help is much appreciated!