For the Yang-Mills field strength defined as a commutator, why does the $A_\nu\partial_\mu - A_\mu\partial_\nu$ term vanish? In basically every QFT book the Yang-Mills strength tensor $F_{\mu\nu}$ is defined as $$F_{\mu\nu}=[D_\mu,D_\nu]$$
where $D_\mu$ is the covariant derivative $$D_\mu=\partial_\mu-A_\mu$$ and $A_\mu$ is the Yang-Mills gauge field.
Explicitly working out the commutator most books obtain (see Peskin 15.15, Srednicki 69.14)
$$F_{\mu\nu}=-\partial_\mu A_\nu +\partial_\nu A_\mu-[A_\mu,A_\nu]$$
However when I work out the commutator I get an extra term $$A_\nu\partial_\mu-A_\mu\partial_\nu$$
This term isn't mentioned in any of the resources I've come across and I don't know what to do with it. Obviously it vanishes somehow. So, 
Question; Why does this term vanish?
 A: Note that, for example,
\begin{align}
[A_\mu,\partial_\nu]f&=A_\mu\partial_\nu f-\partial_\nu(A_\mu f)\\
&=A_\mu\partial_\nu f-\partial_\nu(A_\mu)f-A_\mu\partial_\nu f\\
&=-f\partial_\nu A_\mu\,.
\end{align}
So you don't get terms like $A_\mu\partial_\nu$.
A: This is purely technical issue 
\begin{eqnarray}
F_{\mu\nu} &=& \big[  D_\mu, D_\nu   \big] \;,\\
&=&\big( \partial_\mu-A_\mu    \big)\big( \partial_\nu-A_\nu    \big)-\big( \partial_\nu-A_\nu    \big)\big( \partial_\mu-A_\mu    \big)\;,\\
&=& \partial_\mu \partial_\nu -\partial_\mu A_\nu -A_\mu \partial_\nu + A_\mu A_\nu\\
&& -\partial_\nu \partial_\mu +\partial_\nu A_\mu +A_\nu \partial_\mu - A_\nu A_\mu\;,\\
&=&\big(  \partial_\mu \partial_\nu-\partial_\nu \partial_\mu  \big) -\big(  \partial_\mu A_\nu -\partial_\nu A_\mu \big) -\big( A_\mu \partial_\nu- A_\nu \partial_\mu \big) +\big( A_\mu A_\nu- A_\nu A_\mu  \big)\;,\\
&=&\big(  \partial_\mu \partial_\nu-\partial_\nu \partial_\mu -  A_\mu \partial_\nu+ A_\nu \partial_\mu \big)-\big(  \partial_\mu A_\nu -\partial_\nu A_\mu \big)  +\big( A_\mu A_\nu- A_\nu A_\mu  \big)\;,\\
&=&\big(  D_\mu \partial_\nu-D_\nu \partial_\mu  \big)-\big(  \partial_\mu A_\nu -\partial_\nu A_\mu \big) +\big( A_\mu A_\nu- A_\nu A_\mu  \big)\;,\\
&=& \big(\Gamma_\mu{}^\lambda{}_\nu \partial_\lambda-\Gamma_\nu{}^\beta{}_\mu \partial_\beta \big) -\partial_\mu A_\nu +\partial_\nu A_\mu + \big[A_\mu, A_\nu\big]\;,\\
&=&-\partial_\mu A_\nu +\partial_\nu A_\mu + \big[A_\mu, A_\nu\big]\;.
\end{eqnarray}
Not sure why the last term is flips the sign.
