Here is a really quick derivation of the Cauchy-Riemann equations of complex analysis.
Consider a function of a complex variable, , where , such that:
where and are real-valued functions.
An analytic function is one that is expressible as a power series in .
We formally differentiate this equation as follows. First, differentiating with respect to , we obtain
Differentiating with respect to , we obtain
Multiplying the latter equation by and equating to the first result, we obtain
Comparing imaginary and real parts of these equations, we obtain
which are the famous Cauchy-Riemann equations.