Exercise 1.8 in Stein and Shakarchi's "Complex Analysis" asks you to prove that
$$ \frac{\partial h}{\partial z} = \frac{\partial g}{\partial z} \frac{\partial f}{\partial z} + \frac{\partial g}{\partial \bar{z}} \frac{\partial \bar{f}}{\partial z} $$
where $z = x + iy$, $f$ and $g$ are differentiable in $x$ and $y$ (so not necessarily holomorphic), $h = g \circ f$, $\frac{\partial}{\partial z} = \frac{1}{2} \left( \frac{\partial}{\partial x} + \frac{1}{i} \frac{\partial}{\partial y} \right)$ and $\frac{\partial}{\partial \bar{z}} = \frac{1}{2} \left( \frac{\partial}{\partial x} - \frac{1}{i} \frac{\partial}{\partial y} \right)$.
This immediately seemed wrong to me, because nowhere does it use something like $\frac{\partial g}{\partial f}$. Isn't $f(x + iy) = g(x + iy) = x^2$ a counterexample? Because then $h(x + iy) = x^4$ and we have that $$ \frac{\partial h}{\partial z} = \frac{1}{2}\frac{\partial h}{\partial x} = 2x^3 $$ while for the left hand side I get $$ \frac{\partial g}{\partial z} \frac{\partial f}{\partial z} + \frac{\partial g}{\partial \bar{z}} \frac{\partial \bar{f}}{\partial z} = x \cdot x + x \cdot x = 2x^2 $$ What am I misunderstanding?
