Is the bracket of graded derivations a natural graded derivation?

[MathematicalPhysicist]

I want to prove the next assertion in Jeffrey M. Lee's Manifolds and differential geometry.
If \mathcal{D}_1, \mathcal{D}_2 are (natural) graded derivations of degrees r_1,r_2 respectively, then the operator:
[\mathcal{D}_1,\mathcal{D}_2] := \mathcal{D}_1 \circ \mathcal{D}_2 - (-1)^{r_1 r_2} \mathcal{D}_2 \circ \mathcal{D}_1

is a natural graded derivation of degree r_1+r_2.
I am finding it difficult to prove property 2 and 3 of graded derivation for this bracket.
Property 2 is given in the next page in definition 1.

I am uploading scans of my work (hopefully my hand written work won't stir you away).

 

[mathwonk]

this is a homework type question, i.e. not appropriate.