Complex chain rule for complex valued functions

Let $f=f(z)$ and $g=g(w)$ be two complex valued functions which are differentiable in the real sense, $h(z)=g(f(z))$. Prove the complex chain rule.
All partial derivatives:
$$\frac{\partial h}{\partial z} = \frac{\partial g}{\partial w}\frac{\partial f}{\partial z} + \frac{\partial g}{\partial \bar w}\frac{\partial \bar f}{\partial z}$$
and
$$\frac{\partial h}{\partial \bar z} = \frac{\partial g}{\partial w}\frac{\partial f}{\partial \bar z} + \frac{\partial g}{\partial \bar w}\frac{\partial\bar f}{\partial \bar z}$$
Are we supposed to arrive at this through Cauchy-Riemann?