$F\prime \left(t\right)$

$\=f_x x\prime \left(t\right)\+f_y y\prime \left(t\right)$

$\=2\cos \left(2x-3y\right)\left(1-\frac{1}{t^2}\right)-3\cos \left(2x-3y\right)\left(1+\frac{1}{t^2}\right)$

$\=2\cos \left(t-\frac{5}{t}\right)\left(1-\frac{1}{t^2}\right)-3\cos \left(t-\frac{5}{t}\right)\left(1+\frac{1}{t^2}\right)$

$\= -\cos \left(t-5\/t\right)\left(t^2+5\right)\/t^2$ 

Formal statement of the relevant chain rule

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

Implement the chain rule

$f\left(x\,y\right)\=\sin \left(2 x-3 y\right)$$\stackrel{\text{assign as function}}{\to }$$f$

$X\=t\+1\/t$$\stackrel{\text{assign}}{\to }$

$Y\=t-1\/t$$\stackrel{\text{assign}}{\to }$

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

$2\cos \left(2x-3y\right)\left(1-\frac{1}{t^2}\right)-3\cos \left(2x-3y\right)\left(1+\frac{1}{t^2}\right)$

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

$2\cos \left(t-\frac{5}{t}\right)\left(1-\frac{1}{t^2}\right)-3\cos \left(t-\frac{5}{t}\right)\left(1+\frac{1}{t^2}\right)$

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

$-\frac{\cos \left(\frac{t^2-5}{t}\right)\left(t^2+5\right)}{t^2}$

Obtain $F\prime \left(t\right)$ from the explicit representation $F\left(t\right)\=f\left(x\left(t\right)\,y\left(t\right)\right)$ 

$\frac{\ⅆ}{\ⅆ t} f\left(X\,Y\right)$ = $-\left(1+\frac{5}{t^2}\right)\cos \left(t-\frac{5}{t}\right)$$\stackrel{\text{simplify}}{\=}$$-\frac{\cos \left(\frac{t^2-5}{t}\right)\left(t^2+5\right)}{t^2}$$$

Initialize

$$\begin{gathered} \mathrm{interface}\left(\mathrm{typesetting}\=\mathrm{extended}\right)\: \\ \mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{useprime}\,\mathrm{prime}\=t\right)\: \\ \mathrm{Typesetting}:-\mathrm{Suppress}\left(\left[x\left(t\right)\,y\left(t\right)\right]\right)\: \end{gathered}$$

Formal statement of the relevant chain rule

$\mathrm{diff}\left(f\left(x\left(t\right)\,y\left(t\right)\right)\,t\right)$

${\mathrm{D}}_1\left(f\right)\left(x\,y\right)x\prime +{\mathrm{D}}_2\left(f\right)\left(x\,y\right)y\prime$

Implement the chain rule

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

$f≔\left(x\,y\right)\to \sin \left(2 x-3 y\right)\:$

$$\begin{gathered} X≔t\+1\/t\: \\ Y≔t-1\/t\: \end{gathered}$$$$

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

$-\frac{\cos \left(\frac{t^2-5}{t}\right)\left(t^2+5\right)}{t^2}$

Obtain $F\prime \left(t\right)$ from an explicit representation of $F\left(t\right)$

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

$-\frac{\cos \left(\frac{t^2-5}{t}\right)\left(t^2+5\right)}{t^2}$