I want to show that following $D$ is an exterior derivative. $D:\oplus_i\Omega^i(U)\rightarrow \oplus_i\Omega^i(U)$ is a linear map. Also $D^2=0$, $D(\omega\wedge\tau)=D(\omega)\wedge\tau+(-1)^k\omega\wedge D(\tau)$ for any $w\in \Omega^k(U)$ and $\tau\in\Omega^l(U)$, and $D(f)=df$ for any $f\in\Omega^0(U)$.
The definition that I am using for exterior derivative is from "An Introduction to Manifolds" by Tu.
Then (ii) is satisfied and we need to check (i) and (iii). For (iii) I think we can use $D(f)=df.$ Then (iii) is also verified. However I am not sure how to show (i).
Edit: Meaning of anti derivation of degree 1
