Derivative of complex function.

Question

I intend to compute the $k_{x}-$gradiente of the following function,
$$\phi=\arctan\left(\frac{m}{k_{x}-isk_{y}}\right)$$ where $m\in\mathbb{R}$ and $s=\pm1$. How can I do this?
I think maybe I should use the chain rule where I could start the calculation from,
$$\partial_{x}\phi=\frac{d\phi}{dz}\frac{dz}{dk_{x}}$$ where $z=\dfrac{m}{k_{x}-isk_{y}}=m\dfrac{k_{x}+isk_{y}}{k_{x}^2+(sk_{y})^2}$. But I'm not sure if this is correct.
Thank you in advance.

Answer 1

Thus applying the Chain Rule for the variable $\;k_x\;$:

$$\phi_{k_x}=\frac{-m}{(k_x-isk_y)^2}\cdot\frac1{1+\frac{m^2}{(k_x-isk_y)^2}}=-\frac m{(k_x-isk_y)^2+m^2}$$