For $A \in \mathcal{A}$ viewed as $A[0] \in D^+(\mathcal{A})$, $RF(A[0]) = F(I^\bullet)$ where $A \to I^\bullet$ is an injective resolution. The cohomology $H^n(RF(A[0])) = R^nF(A)$ recovers the classical derived functors. But $RF(A[0])$ is the entire complex, not just its cohomology.

The complex $RF(A)$ contains extension data lost by the individual $R^nF(A)$. For example, the class of $RF(A)$ in $D^+(\mathcal{B})$ determines not just the groups $R^nF(A)$ but also the differentials in spectral sequences computing them. A complex is not determined by its cohomology groups alone.

The total derived functor of $\mathrm{Hom}_R(A, -)$ is $R\mathrm{Hom}_R(A, -) : D^+(R) \to D^+(\mathbf{Ab})$. For complexes $A^\bullet, B^\bullet$, $R\mathrm{Hom}(A^\bullet, B^\bullet)$ is a complex whose cohomology gives $\mathrm{Ext}^n(A^\bullet, B^\bullet) = H^n(R\mathrm{Hom}(A^\bullet, B^\bullet))$.

The total left derived functor of $- \otimes_R -$ is $- \otimes^L_R - : D^-(R) \times D^-(R) \to D^-(R)$. For modules, $A \otimes^L_R B = P_\bullet \otimes_R B$ where $P_\bullet \to A$ is a projective resolution. Then $\mathrm{Tor}_n(A, B) = H^{-n}(A \otimes^L B)$.

$R\Gamma(X, -) : D^+(\mathbf{Sh}(X)) \to D^+(\mathbf{Ab})$ is the total derived functor of $\Gamma(X, -)$. For a sheaf $\mathcal{F}$, $R\Gamma(X, \mathcal{F})$ is a complex whose cohomology is $H^n(X, \mathcal{F})$. One can compute using flasque, soft, or fine resolutions (not just injective).

For $f : X \to Y$, $Rf_* : D^+(\mathbf{Sh}(X)) \to D^+(\mathbf{Sh}(Y))$ is the total derived functor of $f_*$. The stalk formula $(R^nf_*\mathcal{F})_y = H^n(f^{-1}(y), \mathcal{F})$ comes from $H^n$ of the total derived functor. The Leray spectral sequence $H^p(Y, R^qf_*\mathcal{F}) \Rightarrow H^{p+q}(X, \mathcal{F})$ arises from the composition $R\Gamma_Y \circ Rf_* = R\Gamma_X$.

For composable left exact functors $F : \mathcal{A} \to \mathcal{B}$ and $G : \mathcal{B} \to \mathcal{C}$, there is a natural morphism $R(G \circ F) \to RG \circ RF$. The Grothendieck spectral sequence arises from the failure of this to be an isomorphism at the cohomological level: $R^pG(R^qF(A)) \Rightarrow R^{p+q}(G \circ F)(A)$.

Flasque sheaves are $\Gamma$-acyclic, so $R\Gamma(X, \mathcal{F})$ can be computed using a flasque resolution. Similarly, soft sheaves on a paracompact space and fine sheaves on a smooth manifold are $\Gamma$-acyclic. The de Rham complex $\Omega^\bullet_X$ is a fine resolution of $\mathbb{R}_X$, so $R\Gamma(X, \mathbb{R}_X) \simeq \Gamma(X, \Omega^\bullet_X)$.

For $f : X \to Y$, the inverse image $f^* : \mathbf{Sh}(Y) \to \mathbf{Sh}(X)$ is exact, so $Lf^* = f^*$ (no higher derived functors). But for $\mathcal{O}_X$-modules, $f^* = f^{-1}(-) \otimes_{f^{-1}\mathcal{O}_Y} \mathcal{O}_X$ involves a tensor product, giving $Lf^* = f^{-1}(-) \otimes^L_{f^{-1}\mathcal{O}_Y} \mathcal{O}_X$ with higher Tor terms.

If $F \dashv G$ is an adjunction with $F$ right exact and $G$ left exact, then under suitable hypotheses $LF \dashv RG$ is an adjunction at the derived level: $\mathrm{Hom}_{D(\mathcal{B})}(LF(X), Y) \cong \mathrm{Hom}_{D(\mathcal{A})}(X, RG(Y))$. The derived tensor-Hom adjunction $A \otimes^L - \dashv R\mathrm{Hom}(A, -)$ is the prototypical example.

On a scheme $X$, the dualizing complex $\omega_X^\bullet \in D^b_c(X)$ represents the functor $R\mathrm{Hom}(-, \omega_X^\bullet)$, which gives Grothendieck duality: $Rf_* R\mathrm{Hom}(\mathcal{F}, \omega_X^\bullet) \cong R\mathrm{Hom}(Rf_!\mathcal{F}, \omega_Y^\bullet)$. For a smooth variety, $\omega_X^\bullet = \omega_X[\dim X]$.

The functor $R\mathrm{Hom}(\mathcal{E}, -) : D^b(\mathrm{Coh}(X)) \to D^b(\mathrm{Coh}(X))$ for a vector bundle $\mathcal{E}$ is computed using a locally free resolution of $\mathcal{E}$. The Fourier-Mukai transform $\Phi_{\mathcal{P}} = Rp_{2*}(p_1^*(-) \otimes^L \mathcal{P})$ is a composition of total derived functors.

For $f : X \to Y$ proper and $\mathcal{F} \in D^b(X)$, $\mathcal{G} \in D^b(Y)$: the projection formula states $Rf_*(\mathcal{F} \otimes^L f^*\mathcal{G}) \cong Rf_*\mathcal{F} \otimes^L \mathcal{G}$. This is an identity of total derived functors, not just of individual cohomology groups.