= Solution
A function is <holomorphic> on an <open set> if it is <complex differentiable> at every point of that set. At $z_0=x_0+iy_0$, take difference quotients first along real increments. Their limit is $f'(z_0)$, so $u_x=\operatorname{Re}f'(z_0)$ and $v_x=\operatorname{Im}f'(z_0)$. Along imaginary increments,
$$
\frac{f(z_0+ih)-f(z_0)}h\longrightarrow if'(z_0),
$$
so $u_y=-\operatorname{Im}f'(z_0)$ and $v_y=\operatorname{Re}f'(z_0)$. These limits prove existence of all four <partial derivatives> and the <Cauchy-Riemann equations> $u_x=v_y$, $u_y=-v_x$.
For the first example, the identity function and the reciprocal function are <holomorphic> away from zero, so \b[$f$ is <holomorphic> on the given domain], with
$$
\boxed{f'(z)=1-z^{-2}.}
$$
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