I was trying to understand the development of the solution in this answer, where $\overline{f(z)}f'(z)dz = (u - iv)(\frac{\partial u}{\partial x} + i \frac{\partial v}{\partial y})(dx + idy)$. The first term is pretty obvious, as it is just the conjugate. However, where does the equality
$$f'(z)dz = \left(\frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x}\right)(dx + idy)$$ come from?
I know that we can write the derivative of an analytic function $f$ as
$$f'(z) = \frac{\partial u}{\partial x} - i \frac{\partial u}{\partial y} = \frac{\partial v}{\partial y} + i \frac{\partial v}{\partial x}$$
but the above term does not seem to follow algebraically. Where does this term come from?
This comes from the Cauchy-Riemann equations, since $f$ is analytic:
$$\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$$
so it follows from my second expression.