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For a (DF)-space E and a tensor norm α we investigate the derivatives $To{r}_{\alpha }^l(E,·)$ of the tensor product functor $E\otimes {̃}_{\alpha }·:\to$ from the category of Fréchet spaces to the category of linear spaces. Necessary and sufficient conditions for the vanishing of $Tor{¹}_{\alpha }(E,F)$, which is strongly related to the exactness of tensored sequences, are presented and characterizations in the nuclear and (co-)echelon cases are given.