Product Rule for Tensors

$\mathcal{D}(A(X_1,\ldots,X_s)) = \mathcal{D}(C^2 (A \otimes X_1 \otimes X_2)) =$

$C^2(\mathcal{D} (A \otimes X_1 \otimes X_2)) =$

$C^2(((\mathcal{D}A) \otimes X_1 \otimes X_2) + (A \otimes \mathcal{D} X_1 \otimes X_2) + (A \otimes X_1 \otimes \mathcal{D} X_2)) =$

$(\mathcal{D}A)(X_1,X_2) + A(\mathcal{D} X_1,X_2) + A(X_1,\mathcal{D} X_2)$