Let $M$ be a complex manifold. Let $\mathcal{L}$ be a holomorphic line bundle over $M$, with $\pi:M\longrightarrow\mathcal{L}$ being the projection map. Any section $s$ of $\mathcal{L}$ over $M$ is a map $s:M\longrightarrow\mathcal{L}$, with $\pi\circ s=id_M$. Now for any point of $x\in M$, $s(x)\in\mathcal{L}_x\cong\mathbb{C}$ (by fixing an isomorphism). So, we can think of $s$ to be a function $s:M\longrightarrow\mathbb{C}$. Then the corresponding maps between tangent spaces is given by $d_xs:T_xM\longrightarrow T_{s(x)}\mathbb{C}\cong\mathbb{C}$, that is, $d_xs\in T_x^*M$, the cotangent space.
Now if $\mathcal{I}_x$ denotes the sheaf of sections of $\mathcal{L}$ which vanish at $x$, then we get a map $\mathcal{I}_x\longrightarrow T^*_x(M)$ given by $d_x$. Griffiths and Harris says this map is surjective. Why is this?
In particular, how do we get the exact sequence, $0\longrightarrow\tilde{\mathcal{L}}(-2x)\longrightarrow\mathcal{I}_x\longrightarrow T^*_x(M)\otimes k(x)\longrightarrow 0$, where $\tilde{\mathcal{L}}(-2x)$ is the linebundle associated to the divisor $-2x$, and $k(x)$ is the skyscraper sheaf of $\mathcal{L}$ supported at $x$.