I believe this is a computation I have done before, but now I can't write the symbols to convince myself:
What is the connection between the "hard" complex differential operator d/dz and the "soft" $\partial/\partial z$?
How does this make vector fields f(z)d/dz and f(z)$\partial/\partial z$ different?
Thanks in advance for your assistance.
The primary difference is in their connotations. When you write $\frac{\mathrm d}{\mathrm d z}$, I interpret you as meaning that you are working entirely and consistently only with holomorphic (or meromorphic, or analytic, ...) functions, so that you really can treat your domain $\mathbb C$ as one-dimensional. When you write $\frac{\partial}{\partial z}$, I interpret you as allowing functions that have an antiholomorphic part, so that the other partial derivative $\frac\partial{\partial\bar z}$ may also act nontrivially.