Or you could use the reasoning that leads to the Cauchy-Riemann equations. Let z= x+ iy where x and y are real numbers. Then . To find the derivate at, say we form the "difference quotient" and take the limit as z goes . Since that is a two dimensional limit, in order that the derivative exist, the limit must be the same as we approach from any direction.
In particular, what do we get if we approach along the line ?
What do we get if we approach along the line ? Are those the same?