$F_r$

$\=f_x x_r\+f_y y_r\+f_z z_r$

$\=\frac{x{\mathit{\ⅇ}}^r\cos \left(s\right)}{\sqrt{x^2\+y^2\+z^2}}\+\frac{y{\mathit{\ⅇ}}^r\sin \left(s\right)}{\sqrt{x^2\+y^2\+z^2}}\+\frac{zs}{\sqrt{x^2\+y^2\+z^2}}$

$\=\frac{{\mathit{\ⅇ}}^{2 r}{\cos }^2\left(s\right)}{\sqrt{{\mathit{\ⅇ}}^{2 r}{\cos }^2\left(s\right)\+{\mathit{\ⅇ}}^{2 r}{\sin }^2\left(s\right)\+r^2{s}^2}}\+\frac{{\mathit{\ⅇ}}^{2 r}{\sin }^2\left(s\right)}{\sqrt{{\mathit{\ⅇ}}^{2 r}{\cos }^2\left(s\right)\+{\mathit{\ⅇ}}^{2 r}{\sin }^2\left(s\right)\+r^2{s}^2}}\+\frac{r{s}^2}{\sqrt{{\mathit{\ⅇ}}^{2 r}{\cos }^2\left(s\right)\+{\mathit{\ⅇ}}^{2 r}{\sin }^2\left(s\right)\+r^2{s}^2}}$

$\=\frac{{\mathit{\ⅇ}}^{2r}\+r{s}^2}{\sqrt{{\mathit{\ⅇ}}^{2r}\+r^2{s}^2}}$

$F_s$

$\=f_x x_s\+f_y y_s\+f_z z_s$

$\=-\frac{x{\mathit{\ⅇ}}^r\sin \left(s\right)}{\sqrt{x^2\+y^2\+z^2}}\+\frac{y{\mathit{\ⅇ}}^r\cos \left(s\right)}{\sqrt{x^2\+y^2\+z^2}}\+\frac{zr}{\sqrt{x^2\+y^2\+z^2}}$

$\=\frac{r^{2}s}{\sqrt{{\mathit{\ⅇ}}^{2r}{\cos }^2\left(s\right)\+{\mathit{\ⅇ}}^{2r}{\sin }^2\left(s\right)\+r^2{s}^2}}$

$\=\frac{r^{2}s}{\sqrt{{\mathit{\ⅇ}}^{2r}\+r^2{s}^2}}$

Formal statement of the relevant chain rules

$f\left(x\left(r\,s\right)\,y\left(r\,s\right)\,z\left(r\,s\right)\right)$$\stackrel{\text{differentiate w.r.t. r}}{\to }$${\mathrm{D}}_1\left(f\right)\left(x\left(r\,s\right)\,y\left(r\,s\right)\,z\left(r\,s\right)\right)\left(\frac{\partial }{\partial r}x\left(r\,s\right)\right)\+{\mathrm{D}}_2\left(f\right)\left(x\left(r\,s\right)\,y\left(r\,s\right)\,z\left(r\,s\right)\right)\left(\frac{\partial }{\partial r}y\left(r\,s\right)\right)\+{\mathrm{D}}_3\left(f\right)\left(x\left(r\,s\right)\,y\left(r\,s\right)\,z\left(r\,s\right)\right)\left(\frac{\partial }{\partial r}z\left(r\,s\right)\right)$

$f\left(x\left(r\,s\right)\,y\left(r\,s\right)\,z\left(r\,s\right)\right)$$\stackrel{\text{differentiate w.r.t. s}}{\to }$${\mathrm{D}}_1\left(f\right)\left(x\left(r\,s\right)\,y\left(r\,s\right)\,z\left(r\,s\right)\right)\left(\frac{\partial }{\partial s}x\left(r\,s\right)\right)\+{\mathrm{D}}_2\left(f\right)\left(x\left(r\,s\right)\,y\left(r\,s\right)\,z\left(r\,s\right)\right)\left(\frac{\partial }{\partial s}y\left(r\,s\right)\right)\+{\mathrm{D}}_3\left(f\right)\left(x\left(r\,s\right)\,y\left(r\,s\right)\,z\left(r\,s\right)\right)\left(\frac{\partial }{\partial s}z\left(r\,s\right)\right)$

Implement the chain rule

$f\left(x\,y\,z\right)\=\sqrt{x^2\+y^2\+z^2}$$\stackrel{\text{assign as function}}{\to }$$f$$$

$X\={\ⅇ}^r \cos \left(s\right)$$\stackrel{\text{assign}}{\to }$$$

$Y\={\ⅇ}^r \sin \left(s\right)$$\stackrel{\text{assign}}{\to }$$$

$Z\=r s$$\stackrel{\text{assign}}{\to }$

$\left(\frac{\partial }{\partial x} f\left(x\,y\,z\right)\right) \left(\frac{\ⅆ}{\ⅆ r} X\right)\+\left(\frac{\partial }{\partial y} f\left(x\,y\,z\right)\right) \left(\frac{\ⅆ}{\ⅆ r} Y\right)\+\left(\frac{\partial }{\partial z} f\left(x\,y\,z\right)\right) \left(\frac{\ⅆ}{\ⅆ r} Z\right)$

$\frac{x{\ⅇ}^r\cos \left(s\right)}{\sqrt{x^2\+y^2\+z^2}}\+\frac{y{\ⅇ}^r\sin \left(s\right)}{\sqrt{x^2\+y^2\+z^2}}\+\frac{zs}{\sqrt{x^2\+y^2\+z^2}}$

$\stackrel{\text{evaluate at point}}{\to }$

$\frac{{\left({\ⅇ}^r\right)}^2{\cos \left(s\right)}^2}{\sqrt{{\left({\ⅇ}^r\right)}^2{\cos \left(s\right)}^2\+{\left({\ⅇ}^r\right)}^2{\sin \left(s\right)}^2\+r^2{s}^2}}\+\frac{{\left({\ⅇ}^r\right)}^2{\sin \left(s\right)}^2}{\sqrt{{\left({\ⅇ}^r\right)}^2{\cos \left(s\right)}^2\+{\left({\ⅇ}^r\right)}^2{\sin \left(s\right)}^2\+r^2{s}^2}}\+\frac{r{s}^2}{\sqrt{{\left({\ⅇ}^r\right)}^2{\cos \left(s\right)}^2\+{\left({\ⅇ}^r\right)}^2{\sin \left(s\right)}^2\+r^2{s}^2}}$

$\stackrel{\text{simplify}}{\=}$

$\frac{{\ⅇ}^{2r}\+r{s}^2}{\sqrt{{\ⅇ}^{2r}\+r^2{s}^2}}$

$\left(\frac{\partial }{\partial x} f\left(x\,y\,z\right)\right) \left(\frac{\ⅆ}{\ⅆ s} X\right)\+\left(\frac{\partial }{\partial y} f\left(x\,y\,z\right)\right) \left(\frac{\ⅆ}{\ⅆ s} Y\right)\+\left(\frac{\partial }{\partial z} f\left(x\,y\,z\right)\right) \left(\frac{\ⅆ}{\ⅆ s} Z\right)$

$-\frac{x{\ⅇ}^r\sin \left(s\right)}{\sqrt{x^2\+y^2\+z^2}}\+\frac{y{\ⅇ}^r\cos \left(s\right)}{\sqrt{x^2\+y^2\+z^2}}\+\frac{zr}{\sqrt{x^2\+y^2\+z^2}}$

$\stackrel{\text{evaluate at point}}{\to }$

$\frac{r^{2}s}{\sqrt{{\left({\ⅇ}^r\right)}^2{\cos \left(s\right)}^2\+{\left({\ⅇ}^r\right)}^2{\sin \left(s\right)}^2\+r^2{s}^2}}$

$\stackrel{\text{simplify}}{\=}$

$\frac{r^{2}s}{\sqrt{{\ⅇ}^{2r}\+r^2{s}^2}}$

Obtain $F_r$ and $F_s$ from the explicit representation $F\left(r\,s\right)\=f\left(x\left(r\,s\right)\,y\left(r\,s\right)\,z\left(r\,s\right)\right)$ 

$\frac{\partial }{\partial r} f\left(X\,Y\,Z\right)$ = $\frac{1}{2}\frac{2{\left({\ⅇ}^r\right)}^2{\cos \left(s\right)}^2\+2{\left({\ⅇ}^r\right)}^2{\sin \left(s\right)}^2\+2r{s}^2}{\sqrt{{\left({\ⅇ}^r\right)}^2{\cos \left(s\right)}^2\+{\left({\ⅇ}^r\right)}^2{\sin \left(s\right)}^2\+r^2{s}^2}}$$\stackrel{\text{simplify}}{\=}$$\frac{{\ⅇ}^{2r}\+r{s}^2}{\sqrt{{\ⅇ}^{2r}\+r^2{s}^2}}$ $$$$

$\frac{\partial }{\partial s} f\left(X\,Y\,Z\right)$ = $\frac{r^{2}s}{\sqrt{{\left({\ⅇ}^r\right)}^2{\cos \left(s\right)}^2\+{\left({\ⅇ}^r\right)}^2{\sin \left(s\right)}^2\+r^2{s}^2}}$$\stackrel{\text{simplify}}{\=}$$\frac{r^{2}s}{\sqrt{{\ⅇ}^{2r}\+r^2{s}^2}}$$$$$

Initialize

$$\begin{gathered} \mathrm{interface}\left(\mathrm{typesetting}\=\mathrm{extended}\right)\: \\ \mathrm{Typesetting}:-\mathrm{Suppress}\left(\left[x\left(r\,s\right)\,y\left(r\,s\right)\,z\left(r\,s\right)\right]\right)\: \\ \mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}\=\mathrm{true}\right)\: \end{gathered}$$

Formal statement of the relevant chain rules

$\mathrm{diff}\left(f\left(x\left(r\,s\right)\,y\left(r\,s\right)\,z\left(r\,s\right)\right)\,r\right)$

${\mathrm{D}}_1\left(f\right)\left(x\,y\,z\right)\left(x_r\right)+{\mathrm{D}}_2\left(f\right)\left(x\,y\,z\right)\left(y_r\right)+{\mathrm{D}}_3\left(f\right)\left(x\,y\,z\right)\left(z_r\right)$

$\mathrm{diff}\left(f\left(x\left(r\,s\right)\,y\left(r\,s\right)\,z\left(r\,s\right)\right)\,s\right)$

${\mathrm{D}}_1\left(f\right)\left(x\,y\,z\right)\left(x_s\right)+{\mathrm{D}}_2\left(f\right)\left(x\,y\,z\right)\left(y_s\right)+{\mathrm{D}}_3\left(f\right)\left(x\,y\,z\right)\left(z_s\right)$

Implement the chain rule

$\mathrm{Typesetting}:-\mathrm{Unsuppress}\left(\left[x\left(r\,s\right)\,y\left(r\,s\right)\,z\left(r\,s\right)\right]\right)\:$

$f≔\left(x\,y\,z\right)\to \sqrt{x^2\+y^2\+z^2}\:$

$$\begin{gathered} X≔{\ⅇ}^r \cos \left(s\right)\: \\ Y≔{\ⅇ}^r \sin \left(s\right)\: \end{gathered}$$$Z≔r s\:$

$\mathrm{simplify}\left({\mathrm{D}}_1\left(f\right)\left(X\,Y\,Z\right) \mathrm{diff}\left(X\,r\right)\+{\mathrm{D}}_2\left(f\right)\left(X\,Y\,Z\right) \mathrm{diff}\left(Y\,r\right)\+{\mathrm{D}}_3\left(f\right)\left(X\,Y\,Z\right) \mathrm{diff}\left(Z\,r\right)\right)$

$\frac{{\ⅇ}^{2r}+r{s}^2}{\sqrt{{\ⅇ}^{2r}+r^2{s}^2}}$

$\mathrm{simplify}\left({\mathrm{D}}_1\left(f\right)\left(X\,Y\,Z\right) \mathrm{diff}\left(X\,s\right)\+{\mathrm{D}}_2\left(f\right)\left(X\,Y\,Z\right) \mathrm{diff}\left(Y\,s\right)\+{\mathrm{D}}_3\left(f\right)\left(X\,Y\,Z\right) \mathrm{diff}\left(Z\,s\right)\right)$

$\frac{r^{2}s}{\sqrt{{\ⅇ}^{2r}+r^2{s}^2}}$

Obtain $F_r$ and $F_s$ from an explicit representation of $F\left(r\,s\right)$

$\mathrm{simplify}\left(\mathrm{diff}\left(f\left(X\,Y\,Z\right)\,r\right)\right)$

$\frac{{\ⅇ}^{2r}+r{s}^2}{\sqrt{{\ⅇ}^{2r}+r^2{s}^2}}$

$\mathrm{simplify}\left(\mathrm{diff}\left(f\left(X\,Y\,Z\right)\,s\right)\right)$

$\frac{r^{2}s}{\sqrt{{\ⅇ}^{2r}+r^2{s}^2}}$