= Solution
At every $z=x+iy\in U$, the <Cauchy-Riemann equations> are
$$
\boxed{u_x=v_y,\qquad u_y=-v_x}.
$$
To derive them from <complex differentiability>, approach $z$ along real increments $h$. The <complex derivative> gives
$$
f'(z)=\lim_{h\to0}\frac{f(z+h)-f(z)}h=u_x+iv_x.
$$
Approaching along imaginary increments $ih$ instead gives
$$
f'(z)=\lim_{h\to0}\frac{f(z+ih)-f(z)}{ih}=v_y-iu_y.
$$
These limits exist because $f$ is <holomorphic>. Equating their real and imaginary parts proves the equations, together with $f'=u_x+iv_x=v_y-iu_y$.
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