I'm trying to prove the next:
Let $\mathcal{D}$ be a tensor derivation on $M.$ Relative to a coordinate system,
a) if $\mathcal{D}(\partial_{i})=\sum F_{i}^{j}\partial_{j},$ show that $\mathcal{D}(dx^{j})=-\sum F_{i}^{j}dx^{i}.$
b) If $A$ is a $(1,2)$ tensor field, find a formula for the components of $\mathcal{D}A$ in terms of $F_{i}^{j}$ and the components of $A.$
To prove a) we can use that, for a one-form, $(\mathcal{D}\theta)(X)=\mathcal{D}(\theta X)-\theta(\mathcal{D}X).$ Then $$\mathcal{D}(dx^{i})(\partial_{j})=\mathcal{D}(dx^{i}(\partial_{j}))-dx^{i}(\mathcal{D}\partial_{j})=-dx^{i}(\sum F_{j}^{k}\partial_{k})=-F_{j}^{i}.$$
Then $\mathcal{D}(dx^{i})=-\sum F_{j}^{i}dx^{j}.$
For part b) I'm stuck. Could you give me a hand with this?
Any kind of help is thanked in advanced.