= Solution
Write $f=u+iv$ and $z=x+iy$. The <complex derivative> computed along a real increment is $f'=u_x+iv_x$, while along an imaginary increment it is $f'=v_y-iu_y$. Equality gives
$$
\boxed{u_x=v_y,\qquad u_y=-v_x.}
$$
These are the <Cauchy-Riemann equations>[Cauchy–Riemann equations] at every point where $f$ is <holomorphic>. The directional calculations also establish the existence of those <partial derivatives>. More explicitly, $f(z+h)-f(z)=f'(z)h+o(|h|)$ gives both real-direction formulas directly. If a converse is desired, <continuous> first <partial derivatives> together with these equations make the real differential multiplication by $u_x+iv_x$, hence give <complex differentiability>; the necessity above is what is requested.
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