I am trying to calculate $$\mathcal{\bar{D}}_\dot{\alpha}y^\mu= \left(\bar{\partial_\dot{\alpha}}+i\theta^\alpha\sigma^\mu_{\alpha\dot{\alpha}}\partial_\mu\right)\left(x^\mu+i\theta^\beta\sigma^\mu_{\beta\dot{\beta}}\bar{\theta}^\dot{\beta} \right)$$ $$ =i\theta^\beta\sigma^\mu_{\beta\dot{\beta}}\bar{\partial}_\dot{\alpha}\bar{\theta}^\dot{\beta}+i\theta^\alpha\sigma^\mu_{\alpha\dot{\alpha}}\bar{\theta}^\dot{\alpha}\partial_\mu x^\mu $$ $$ =2i(\theta\sigma^\mu)_\dot{\alpha} $$
$y^\mu$ is a transformation from the $x^\mu$ coordinate. I should have gotten $\mathcal{\bar{D}}_\dot{\alpha}y^\mu=0$ as mentioned in the notes. Is one of the 2 terms on the 2nd line negative? That would cancel out.
I suspect my mistake might be with the grassmann properties(missed a - sign).
Source: (Bertolinis Notes on SUSY) (section 4.4)
