$u_x^2\+u_y^2\+ u_z^2$

$\={\left(g\prime r_x\right)}^2\+{\left(g\prime r_y\right)}^2\+{\left(g\prime r_z\right)}^2$

$\={\left(g\prime \frac{x}{r}\right)}^2\+{\left(g\prime \frac{y}{r}\right)}^2\+{\left(g\prime \frac{z}{r}\right)}^2$

$\={\left(g\prime \right)}^2\left(\frac{x^2\+y^2\+z^2}{r^2}\right)$

$\={\left(g\prime \right)}^2\left(\frac{x^2\+y^2\+z^2}{x^2\+y^2\+z^2}\right)$

$\={\left(g\prime \right)}^2$

$\={\left(\frac{\mathrm{du}}{\mathrm{dr}}\right)}^2$

Define $r$ and $u$ 

$r\=\sqrt{x^2\+y^2\+z^2}$$\stackrel{\text{assign}}{\to }$

$u\=g\left(r\right)$$\stackrel{\text{assign}}{\to }$

Compute $u_x^2\+u_y^2\+ u_z^2$ 

${\left(\frac{\partial }{\partial x} u\right)}^2\+ {\left(\frac{\partial }{\partial y} u\right)}^2\+{\left(\frac{\partial }{\partial z} u\right)}^2$ = $\frac{{\mathrm{D}\left(g\right)\left(\sqrt{x^2\+y^2\+z^2}\right)}^2{x}^2}{x^2\+y^2\+z^2}\+\frac{{\mathrm{D}\left(g\right)\left(\sqrt{x^2\+y^2\+z^2}\right)}^2{y}^2}{x^2\+y^2\+z^2}\+\frac{{\mathrm{D}\left(g\right)\left(\sqrt{x^2\+y^2\+z^2}\right)}^2{z}^2}{x^2\+y^2\+z^2}$$\stackrel{\text{simplify}}{\=}$${\mathrm{D}\left(g\right)\left(\sqrt{x^2\+y^2\+z^2}\right)}^2$

Obtain and square the partial derivatives $u_x\,u_y$, and $u_z$ 

$\frac{\partial }{\partial x} u$ = $\frac{\mathrm{D}\left(g\right)\left(\sqrt{x^2\+y^2\+z^2}\right)x}{\sqrt{x^2\+y^2\+z^2}}$$$

${\left(\frac{\partial }{\partial x} u\right)}^2$ = $\frac{{\mathrm{D}\left(g\right)\left(\sqrt{x^2\+y^2\+z^2}\right)}^2{x}^2}{x^2\+y^2\+z^2}$$$

$\frac{\partial }{\partial y} u$ = $\frac{\mathrm{D}\left(g\right)\left(\sqrt{x^2\+y^2\+z^2}\right)y}{\sqrt{x^2\+y^2\+z^2}}$$$

${\left(\frac{\partial }{\partial y} u\right)}^2$ = $\frac{{\mathrm{D}\left(g\right)\left(\sqrt{x^2\+y^2\+z^2}\right)}^2{y}^2}{x^2\+y^2\+z^2}$$$

$\frac{\partial }{\partial z} u$ = $\frac{\mathrm{D}\left(g\right)\left(\sqrt{x^2\+y^2\+z^2}\right)z}{\sqrt{x^2\+y^2\+z^2}}$$$

${\left(\frac{\partial }{\partial z} u\right)}^2$ = $\frac{{\mathrm{D}\left(g\right)\left(\sqrt{x^2\+y^2\+z^2}\right)}^2{z}^2}{x^2\+y^2\+z^2}$$$

Initialize

$r≔\sqrt{x^2\+y^2\+z^2}\:$

$u≔g\left(r\right)\:$

Apply the diff and simplify commands to evaluate the expression $u_x^2\+u_y^2\+ u_z^2$ 

$\mathrm{simplify}\left(\mathrm{diff}{\left(u\,x\right)}^2\+ \mathrm{diff}{\left(u\,y\right)}^2\+\mathrm{diff}{\left(u\,z\right)}^2\right)$ = ${\mathrm{D}\left(g\right)\left(\sqrt{x^2\+y^2\+z^2}\right)}^2$$$

Separately obtain $u_x\,u_y$, and $u_z$ and their squares

$\mathrm{diff}\left(u\,x\right)$ = $\frac{\mathrm{D}\left(g\right)\left(\sqrt{x^2\+y^2\+z^2}\right)x}{\sqrt{x^2\+y^2\+z^2}}$$$$$

$\mathrm{diff}{\left(u\,x\right)}^2$ = $\frac{{\mathrm{D}\left(g\right)\left(\sqrt{x^2\+y^2\+z^2}\right)}^2{x}^2}{x^2\+y^2\+z^2}$$$

$\mathrm{diff}\left(u\,y\right)$ = $\frac{\mathrm{D}\left(g\right)\left(\sqrt{x^2\+y^2\+z^2}\right)y}{\sqrt{x^2\+y^2\+z^2}}$$$$$

$\mathrm{diff}{\left(u\,y\right)}^2$ = $\frac{{\mathrm{D}\left(g\right)\left(\sqrt{x^2\+y^2\+z^2}\right)}^2{y}^2}{x^2\+y^2\+z^2}$$$

$\mathrm{diff}\left(u\,z\right)$ = $\frac{\mathrm{D}\left(g\right)\left(\sqrt{x^2\+y^2\+z^2}\right)z}{\sqrt{x^2\+y^2\+z^2}}$$$$$

$\mathrm{diff}{\left(u\,z\right)}^2$ = $\frac{{\mathrm{D}\left(g\right)\left(\sqrt{x^2\+y^2\+z^2}\right)}^2{z}^2}{x^2\+y^2\+z^2}$$$