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This article is about the derivation algebra of multi-loop
algebras. Multi-loop algebras are algebras obtained by a
generalization of a process known as twisting by automorphisms in
the theory of Kac-Moody algebras. Multi-loop algebras are used in
the realization of extended affine Lie algebras. Under certain
conditions on an algebra $\mathcal{A}$, we determine the
derivation algebra of an $n$-step multi-loop algebra based on
$\mathcal{A}$ as the semidirect product of a multi-loop algebra
based on the derivation algebra of $\mathcal{A}$ and the
derivation algebra of the Laurent polynomials in $n$-variables.
This in particular determines the derivation algebras of the core
modulo center of (almost all) extended affine Lie algebras.
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