This may be quite a petty question, but it's been bothering me for a while.
So for the vector fields $X$ and $Y$, we can write them in component form as
$X=X^{\mu}\frac{\partial}{\partial x^{\mu}}\space \space \space $ and $\space \space \space Y=Y^{\nu}\frac{\partial}{\partial x^{\nu}}$
(and we'll abbreviate these as $X=X^{\mu}\partial_{\mu}$ and $Y=Y^{\nu}\partial_{\nu}$).
In finding $[X,Y]$ it seems to suffice to show
$[X,Y]=XY-YX=X(Y^{\mu}\partial_{\mu})-Y(X^{\mu}\partial_{\mu})=(X^{\nu}\partial_{\nu}Y^{\mu}-Y^{\nu}\partial_{\nu}X^{\mu})\partial_{\mu}$ $\space \space \space \space$ (1)
However, does this work:
$[X,Y]=XY-YX=X^{\nu}\partial_{\nu}(Y^{\mu}\partial_{\mu})-Y^{\nu}\partial_{\nu}(X^{\mu}\partial_{\mu})$
then apply the product rule to get
$X^{\nu}\partial_{\nu}(Y^{\mu})\partial_{\mu}+X^{\nu}Y^{\mu}\partial_{\nu}\partial_{\mu}-Y^{\nu}\partial_{\nu}(X^{\mu})\partial_{\mu}-Y^{\nu}X^{\mu}\partial_{\nu}\partial_{\mu}$ $\space \space \space \space$ (2)
I can't see why this isn't formally correct? And if so, the second and the fourth term cancel, or are they both zero? Or is (1) just a simplified version of (2) and if fact none of the terms are zero nor cancel. And why?