<s>[INST] What is $\frac{{\partial f(\left| z \right|)}}{{\partial x}} = $? Let $f(z)$ is polynomial and $z=x+iy$ , $x,y\in R$.
What is $\frac{{\partial f(\left| z \right|)}}{{\partial x}} = $?
 [/INST] $$\begin{align} & \frac{{\partial f(\left| z \right|)}}{{\partial x}}
\\ & =\frac{{\partial f(\sqrt{x^2+y^2})}}{{\partial x}}
\\ & =\frac{\partial f(\sqrt{x^2+y^2})}{\partial (\sqrt{x^2+y^2})}\cdot \frac{\partial (\sqrt{x^2+y^2})}{\partial x}
\\ & =\frac{d\{f(\sqrt{x^2+y^2})\}}{d(\sqrt{x^2+y^2})}\cdot \frac{\partial (\sqrt{x^2+y^2})}{\partial x}
\\ & =\color{red}{f'(\sqrt{x^2+y^2})\cdot \frac{x}{\sqrt{x^2+y^2}}}\end{align}$$
Hope this helps.
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