[Math] Prove that if $f$ and $g$ are analytic at $w$, then so is $fg$

Question

Prove that if $f$ and $g$ are analytic at $w$, then so is $fg.$

My main attempt was using the Cauchy-Riemann equations on the product in this manner but this did not work out. My thinking:

\begin{align}
f =u_1+iv_1, \\
g =u_2+iv_2,
\end{align}

then $fg=(u_1u_2-v_1v_2)+i(u_1v_2+v_1u_2)$.
Set $\phi= u_1u_2-v_1v_2, \ \pi = u_1v_2+v_1u_2$.
I need to prove
$$
\frac{\partial \phi}{\partial x} = \frac{\partial \pi}{\partial y} \\ \frac{\partial \phi}{\partial y} = -\frac{\partial \pi}{\partial x}
$$

I tried other things as well but nothing is working for me.

Answer 1

Whatever proof you know for $(fg)' = f'g+fg'$ from elementary calculus will work nearly verbatim in this situation.