Deriving the Cauchy-Riemann from the derivative of the complex conjugate.

I am trying to prove the Cauchy-Riemann conditions naturally arise from the condition that:

$$frac{partial f(z,z^{*})}{partial z^{*}} = 0$$

But I’m having trouble starting from the definition of a derivative and get stuck:

$$frac{partial f(z,z^{*})}{partial z^{*}} = lim_{Delta zto 0}frac{f(z,z^{*}_{0}+Delta z)-f(z,z^{*}_{0})}{Delta z}$$

Am I going about this the right way? Is there a better way to do this?

Thanks!