Given an short exact sequence of (coherent) sheaves $0 \to A \xrightarrow f B \xrightarrow g B \to 0$, on a (smooth) projective variety $X$ over an algebraically closed field $k$, one can build a long exact sequence of cohomology groups (that turn out to be just vector spaces in this case)
$0 \to H^0(X,A) \xrightarrow{\Gamma(f)} H^0(X,B)\xrightarrow{\Gamma(g)} H^0(X,C)\xrightarrow{\delta} H^1(X,A) \xrightarrow{R^1\Gamma(f)} H^1(X,B)\xrightarrow{R^1\Gamma(g)} H^1(X,B) \to \cdots$
where $\delta$ is a $\delta$ fuctor (see Hartshorne's Algebraic Geometry book, Chapter III) $\Gamma$ is the global section functor and $R^i(f)$ is the $i$-th derived of $f$.
I would like to know if there is a effective method to compute the $\delta$ functor and $R^i(f)$ (or $R^i(f)$ for some specific cases, for instance when $A,B$ and $C$ are sums of line bundles (hence the maps $f$ and $g$ are just matrices with polynomial entries).
For instante, let $X = \mathbb{P}^n$ and fix an hyperplane $H \subset \mathbb{P}^n$ we have the restriction sequence:
$$0 \to \mathcal{O}_{\mathbb{P}^n}(-1) \xrightarrow{\times H} \mathcal{O}_{\mathbb{P}^n} \to \mathcal{O}_{\mathbb{P}^n}\otimes \mathcal{O}_{H} \to 0$$
Hartshorne says (page 227) that we have the following:
$$H^{n-1}(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}\otimes \mathcal{O}_{H}) \xrightarrow{\delta} H^n(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(-1))\xrightarrow{\times H} H^n(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n})\to 0$$
and that $\delta$ is the "division" by $H$. I would like to understand why is that.
Thank you in advance.
PS: I known that having the matrices $f$ and $g$ for the case where the sheaves are sums of line bundles, I can compute the matrix of the maps in the vector spaces, but I am looking for a more insightfull way to do it.