We determine the abelianizations of the following three kinds of graded Liealgebras in certain stable ranges: derivations of the free associative algebra,derivations of the free Lie algebra and symplectic derivations of the freeassociative algebra. In each case, we consider both the whole derivation Liealgebra and its ideal consisting of derivations with positive degrees. As anapplication of the last case, and by making use of a theorem of Kontsevich, weobtain a new proof of the vanishing theorem of Harer concerning the toprational cohomology group of the mapping class group with respect to itsvirtual cohomological dimension.
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