$\displaystyle u = z^3 \bar{z}^5 + \bar{z}^3 z^5 $

I'm not sure of how to differentiate z z-conjugate. I used $\displaystyle z = x + iy $ and $\displaystyle \bar{z} = x - iy$ and started to differentiate to get $\displaystyle u_x$. When I get to $\displaystyle u_{xx}$ it's very bloated. Is there a neater way of doing this than my way?

To check whether u is harmonic or not I need to check if $\displaystyle \Delta u = u_{xx} + u_{yy} = 0$