Answer
$\left(\frac{\partial w}{\partial \rho}\right)^{2}=\left(\frac{\partial w}{\partial z}\right)^{2}+\left(\frac{\partial w}{\partial y}\right)^{2}+\left(\frac{\partial w}{\partial x}\right)^{2}$
Work Step by Step
Let $f$ be a differential function of one variable
\[
\left(x^{2}+y^{2}+z^{2}\right)^{1 / 2}=\rho
\]
Let $(\rho)f=w$
\[
\begin{array}{l}
\frac{\partial w}{\partial x}=\frac{d w}{d \rho} \frac{\partial \rho}{\partial x} \\
\frac{\partial w}{\partial y}=\frac{d w}{d \rho} \frac{\partial \rho}{\partial y} \\
\frac{\partial w}{\partial z}=\frac{d w}{d \rho} \frac{\partial \rho}{\partial z}
\end{array}
\]
Let's consider $\frac{\partial \rho}{\partial x}=\frac{2 x}{2\left(x^{2}+y^{2}+z^{2}\right)^{1 / 2}}, \quad \frac{\partial \rho}{\partial y}=\frac{2 y}{2\left(x^{2}+y^{2}+z^{2}\right)^{1 / 2}}$
and $\quad \frac{\partial \rho}{\partial z}=\frac{2 z}{2\left(x^{2}+y^{2}+z^{2}\right)^{1 / 2}}$
Thus, here we can write
\[
\left(\frac{\partial w}{\partial x}\right)^{2}+\left(\frac{\partial w}{\partial y}\right)^{2}+\left(\frac{\partial w}{\partial z}\right)^{2}=\left(\frac{\partial w}{\partial \rho}\right)^{2} \frac{x^{2}}{\left(x^{2}+y^{2}+z^{2}\right)}+\left(\frac{\partial w}{\partial \rho}\right)^{2} \frac{y^{2}}{\left(x^{2}+y^{2}+z^{2}\right)}+\left(\frac{\partial w}{\partial \rho}\right)^{2} \frac{z^{2}}{\left(x^{2}+y^{2}+z^{2}\right)}
\]
$\left(\frac{\partial w}{\partial x}\right)^{2}+\left(\frac{\partial w}{\partial y}\right)^{2}+\left(\frac{\partial w}{\partial z}\right)^{2}=\left(\frac{\partial w}{\partial \rho}\right)^{2}\left(\frac{x^{2}+y^{2}+z^{2}}{x^{2}+y^{2}+z^{2}}\right)$
$\left(\frac{\partial w}{\partial \rho}\right)^{2}=\left(\frac{\partial w}{\partial z}\right)^{2}+\left(\frac{\partial w}{\partial y}\right)^{2}+\left(\frac{\partial w}{\partial x}\right)^{2}$
