How to find derivative of this complex function

Question

I have no idea how to differentiate this function with respect to z

I know about cr equation and derivative in forms of that. But i cant apply that.

Answer 1

Hint:

Note that: $$ \frac{\partial w}{\partial x}=\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}\qquad \mbox {and} \qquad \frac{\partial w}{\partial y}=\frac{\partial u}{\partial y}+i\frac{\partial v}{\partial y} $$

and remember that, if $w$ is analytic:

$$ \frac{d w}{d z}=\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}=\frac{\partial v}{\partial y}-i\frac{\partial u}{\partial y} =\frac{\partial u}{\partial x}-i\frac{\partial u}{\partial y}= \frac{\partial v}{\partial y}+i\frac{\partial v}{\partial x} $$