If I have functions $g$ and $f=u+iv$ such that $$h(x,y) = g(u(x,y),v(x,y))$$ is reasonably definded, what would $\partial^2h/\partial x \partial x$ and $\partial^2h/\partial y \partial y$ look like?
For wider context this is part of a problem where I should prove that if $g$ is harmonic and $f$ is holomorphic on reasonable domains then $h$ is harmonic, I have looked at resources online but I think I am making some mistakes when doing the calculations as I can't seem to get anything normal looking.
I suggest using the compact notation that $u_x=\tfrac{\partial u(x,y)}{\partial x}$ and so forth, so that mistakes don't creep into the clutter.
If $g(u,v)$ is a Harmonic function, then $g_{uu}+g_{vv}=0$.
If $u(x,y)+iv(x,y)$ is a Holomorphic function, then $u_x=v_y$ and $v_x=-u_y$. So too do the double derivatives interrelate:
$$u_{yy}=(-v_x)_y=-(v_y)_x=-u_{xx}\\v_{yy}=(u_x)_y=(u_y)_x=-v_{xx}$$
Now, we seek use these properties to show that $h_{xx}+h_{yy}=0$, and therefore demonstrate that $h(x,y)$ is harmonic as well.
So... let's look at those terms.
Take things one step at a time, applying the Chain Rule and Product Rule as needed, then tidying thing up.
$$\begin{align}h_{xx} &=(g_u~u_x+g_v~v_x)_x\tag{Chain Rule}\\&=(g_u)_x~u_x+g_u~u_{xx}+(g_v)_x~v_x+g_v~v_{xx}\tag{Product Rule}\\&~~\vdots\end{align}$$
You should obtain a series of five terms; which are various products of the factors: $g_u, g_{uu}, g_{uv}, g_{vv} , u_x, u_{xx}, v_x$, and $v_{xx}$
Use symmetry, and the above Holomorphic replacements, to express $h_{yy}$ as a similar function of these factors.
Add these together and watch things cancel. $$h_{xx}+h_{yy}=0$$
As required. $\blacksquare$