${\left(w_r\right)}^2-{\left(w_s\/r\right)}^2$

$\={\left(f_x x_r\+f_y y_r\right)}^2-{\left(\left(f_x x_s\+f_y y_s\right)\/r\right)}^2$

$\={\left(f_x \cosh \left(s\right)\+f_y \sinh \left(s\right)\right)}^2-{\left(\left(f_x r \sinh \left(s\right)\+f_y r \cosh \left(s\right)\right)\/r\right)}^2$

$$\begin{gathered} \=\left(f_x^2 {\cosh }^2\left(s\right)\+f_y^2 {\sinh }^2\left(s\right) \+2 {\sinh }^2\left(s\right){\cosh }^2\left(s\right)f_x{f}_y\right) \\ -\left(f_x^2 {\sinh }^2\left(s\right)\+f_y^2 {\cosh }^2\left(s\right) \+2 {\sinh }^2\left(s\right){\cosh }^2\left(s\right)f_x{f}_y\right) \end{gathered}$$

$\=\left({\cosh }^2\left(s\right)-{\sinh }^2\left(s\right)\right)f_x^2-\left({\cosh }^2\left(s\right)-{\sinh }^2\left(s\right)\right)f_y^2$

$\=1\cdot f_x^2-1\cdot f_y^2$

$\=f_x^2-f_y^2$

Define $r\,s$, and $u$ 

$x\=r \cosh \left(s\right)$$\stackrel{\text{assign}}{\to }$

$y\=r \sinh \left(s\right)$$\stackrel{\text{assign}}{\to }$

$w\=f\left(x\,y\right)$$\stackrel{\text{assign}}{\to }$

Compute ${\left(w_r\right)}^2-{\left(w_s\/r\right)}^2$ and simplify the result

${\left(\frac{\partial }{\partial r} w\right)}^2-{\left(\left(\frac{\partial }{\partial s} w\right)\/r\right)}^2$ = ${\left({\mathrm{D}}_1\left(f\right)\left(r\cosh \left(s\right)\,r\sinh \left(s\right)\right)\cosh \left(s\right)\+{\mathrm{D}}_2\left(f\right)\left(r\cosh \left(s\right)\,r\sinh \left(s\right)\right)\sinh \left(s\right)\right)}^2-\frac{{\left({\mathrm{D}}_1\left(f\right)\left(r\cosh \left(s\right)\,r\sinh \left(s\right)\right)r\sinh \left(s\right)\+{\mathrm{D}}_2\left(f\right)\left(r\cosh \left(s\right)\,r\sinh \left(s\right)\right)r\cosh \left(s\right)\right)}^2}{r^2}$$\stackrel{\text{simplify}}{\=}$${{\mathrm{D}}_1\left(f\right)\left(r\cosh \left(s\right)\,r\sinh \left(s\right)\right)}^2-{{\mathrm{D}}_2\left(f\right)\left(r\cosh \left(s\right)\,r\sinh \left(s\right)\right)}^2$

Separately obtain and simplify the partial derivatives $u_x\,u_y$, and $u_z$ 

$\frac{\partial }{\partial r} w$ = ${\mathrm{D}}_1\left(f\right)\left(r\cosh \left(s\right)\,r\sinh \left(s\right)\right)\cosh \left(s\right)\+{\mathrm{D}}_2\left(f\right)\left(r\cosh \left(s\right)\,r\sinh \left(s\right)\right)\sinh \left(s\right)$

$\frac{\partial }{\partial s} w$ = ${\mathrm{D}}_1\left(f\right)\left(r\cosh \left(s\right)\,r\sinh \left(s\right)\right)r\sinh \left(s\right)\+{\mathrm{D}}_2\left(f\right)\left(r\cosh \left(s\right)\,r\sinh \left(s\right)\right)r\cosh \left(s\right)$

Obtain $x_r\,x_s$ and $y_r\,y_s$ 

$\frac{\partial }{\partial r} x$ = $\cosh \left(s\right)$ 

$\frac{\partial }{\partial r} y$ = $\sinh \left(s\right)$ 

$\frac{\partial }{\partial s} x$ = $r\sinh \left(s\right)$ 

$\frac{\partial }{\partial s} y$ = $r\cosh \left(s\right)$ 

Initialize

$x≔r \cosh \left(s\right)\:$

$y≔r \sinh \left(s\right)\:$

$w≔f\left(x\,y\right)\:$

Apply the diff and simplify commands to evaluate the expression ${\left(w_r\right)}^2-{\left(w_s\/r\right)}^2$ 

$\mathrm{simplify}\left(\mathrm{diff}{\left(w\,r\right)}^2-{\left( \mathrm{diff}\left(w\,s\right)\/r\right)}^2\right)$ = ${{\mathrm{D}}_1\left(f\right)\left(r\cosh \left(s\right)\,r\sinh \left(s\right)\right)}^2-{{\mathrm{D}}_2\left(f\right)\left(r\cosh \left(s\right)\,r\sinh \left(s\right)\right)}^2$

Obtain $w_r$ and $w_s$ 

$\mathrm{diff}\left(w\,r\right)$ = ${\mathrm{D}}_1\left(f\right)\left(r\cosh \left(s\right)\,r\sinh \left(s\right)\right)\cosh \left(s\right)\+{\mathrm{D}}_2\left(f\right)\left(r\cosh \left(s\right)\,r\sinh \left(s\right)\right)\sinh \left(s\right)$

$\mathrm{diff}\left(w\,s\right)$ = ${\mathrm{D}}_1\left(f\right)\left(r\cosh \left(s\right)\,r\sinh \left(s\right)\right)r\sinh \left(s\right)\+{\mathrm{D}}_2\left(f\right)\left(r\cosh \left(s\right)\,r\sinh \left(s\right)\right)r\cosh \left(s\right)$

Obtain the derivatives $x_r\,x_s$ and $y_r\,y_s$ 

$\mathrm{diff}\left(x\,r\right)$ = $\cosh \left(s\right)$

$\mathrm{diff}\left(y\,r\right)$ = $\sinh \left(s\right)$

$\mathrm{diff}\left(x\,s\right)$ = $r\sinh \left(s\right)$

$\mathrm{diff}\left(y\,s\right)$ = $r\cosh \left(s\right)$