I heard many times that because of introducing derived category, we can avoid cumbersome spectral sequence. However, I don't quite understand its meaning. Here is a precise example people talking about:
Let $f: X \to Y, g: Y \to Z$ be morphism of smooth varieties, and Let $D(X), D(Y), D(Z)$ be derived category of bounded complexes of quasi-coherent sheaves. Then it is said that because of the associativity of derived functor, i.e.
$${\rm{R}}f_* \circ {\rm{R}}g_* = {\rm{R}}{(f \circ g)}_*\quad,$$
we can avoid the write the spectral sequence.
I don't know which spectral sequence people refer to, and why the associativity of derived functor equivalent to that spectral sequence?
The spectral sequence is the Leray spectral sequence $$E_2^{i,j} := R^if_* R^jg_* \mathcal F\implies R^{i+j}(f\circ g)_*\mathcal F.$$