Answer
$$\frac{\partial w}{\partial x}=\frac{x}{\sqrt{x^2+y^2+z^2}}$$
$$\frac{\partial w}{\partial y}=\frac{y}{\sqrt{x^2+y^2+z^2}}$$
$$\frac{\partial w}{\partial z}=\frac{z}{\sqrt{x^2+y^2+z^2}}$$
Work Step by Step
We will use chain rule to find these partial derivatives.
The partial derivative with respect to $x$ is:
$$\frac{\partial w}{\partial x}=\frac{\partial }{\partial x}(\sqrt{x^2+y^2+z^2})=\frac{1}{2\sqrt{x^2+y^2+z^2}}\frac{\partial }{\partial x}(x^2+y^2+z^2)=\frac{1}{2\sqrt{x^2+y^2+z^2}}\cdot2x=\frac{x}{\sqrt{x^2+y^2+z^2}}$$
We treated $y$ and $z$ as a constant, as we did with $y^2$ and $z^2$.
The partial derivatives with respect to $y$ is:
$$\frac{\partial w}{\partial y}=\frac{\partial }{\partial y}(\sqrt{x^2+y^2+z^2})=\frac{1}{2\sqrt{x^2+y^2+z^2}}\frac{\partial }{\partial y}(x^2+y^2+z^2)=\frac{1}{2\sqrt{x^2+y^2+z^2}}\cdot2y=\frac{y}{\sqrt{x^2+y^2+z^2}}$$
We treated $x$ and $z$ as a constant, as we did with $x^2$ and $z^2$.
The partial derivative with respect to $z$ is:
$$\frac{\partial w}{\partial z}=\frac{\partial }{\partial z}(\sqrt{x^2+y^2+z^2})=\frac{1}{2\sqrt{x^2+y^2+z^2}}\frac{\partial }{\partial z}(x^2+y^2+z^2)=\frac{1}{2\sqrt{x^2+y^2+z^2}}\cdot2z=\frac{z}{\sqrt{x^2+y^2+z^2}}$$
We treated $x$ and $y$ as a constant, as we did with $x^2$ and $y^2$.
