For the derivative of the complex function, when it is analytic, then it satisfies Cauchy-Riemann equation. So in this case we have $f'(z_{0})=u_{x}(x_{0},y_{0})+iv_{x}(x_{0},y_{0})$. But if we consider the Vector valued function $g:\mathbb{R^{2}}\rightarrow\mathbb{R^{2}}$, the derivative at $z_{0}=(x_{0},y_{0})$ is $g'(z_{0})$, which is a Jacobi matrix. What makes these two things different? Is because the binary operation "vector multiplication" makes different sense in complex plane and $\mathbb{R^2}$?
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