Show that if φ(x, y) is harmonic in a domain D, then f(z) = φ$\\_{x}$(x, y) − iφ$\\_{y}$(x, y) is analytic in D.
I figure that φ$\\_{xx}$ = - φ$\\_{yy}$ from "φ(x, y) is harmonic" so then that means $\frac{d}{dx}$ (φ$\\_{x}($x, y))= φ$\\_{xx}$ is equal to $\frac{d}{dy}$ (-φ$\\_{y}$(x, y))= -φ$\\_{yy}$ f$\\_{x} $=-f$\\_{y}$ is satisfied.
What I don't know however is how to arrive at φ$\\_{xy}$ = φ$\\_{yx}$ (the other Cauchy Riemann equation). I have a feeling it has something to do with the chain rule but I'm not sure what to do.
partial derivatives commute, this is known as Schwartz's or Clairaut's theorem.