I'm a beginner in Sagemath and I search for help with a calculation problem with some derivatives and basic operations.

I need to simplify a PDE that depends of a smooth function $f:U\subset\mathbb{C}\to\mathbb{C}$, something like, $$Af^2+2fB+C=0\tag{1}$$ where $A$, $B$ and $C$ depends of $f$ and $\bar{f}$ and their complex derivatives, of order until 3.

Of course I can compute this by hand, but in my problem it will be a very long computation and I thought about to try Sagemath for this computation.

I would like to know if I can to define a symbolic complex function on Sagemath and to do basic operations as to compute derivatives in $z$ and $\bar{z}$. For example, if

$A=(f+\bar{f}_{z}f_{z\bar{z}})_z$

$B=(f^2+\bar{f}_{z}^2f_{z\bar{z}}^2)_z$

$C=(f^3+\bar{f}_{z}^3f_{z\bar{z}}^3)_z,$

Can I simplify equation $(1)$ using Sagemath?

Thank for your attention. I appreciate any help.