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

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

$\=\left(-2 x\right)\cdot 1\+\left(-2 y\right)\cdot \left(2 t\right)$

$\=\left(-2 t\right)\cdot 1\+\left(-2 t^2\right)\cdot \left(2 t\right)$

$\=-2 t-4 t^3$

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)\=3-x^2-y^2$$\stackrel{\text{assign as function}}{\to }$$f$

$\left(\frac{\partial }{\partial x} f\left(x\,y\right)\right) \left(\frac{\ⅆ}{\ⅆ t} t\right)\+\left(\frac{\partial }{\partial y} f\left(x\,y\right)\right) \left(\frac{\ⅆ}{\ⅆ t} t^2\right)$ = $-4ty-2x$$\stackrel{\text{evaluate at point}}{\to }$$-4t^3-2t$

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

$f\left(t\,t^2\right)$

$-t^4-t^2\+3$

$\stackrel{\text{differentiate w.r.t. t}}{\to }$

$-4t^3-2t$

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

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

$f≔\left(x\,y\right)\to 3-x^2-y^2\:$

${\mathrm{D}}_1\left(f\right)\left(t\,t^2\right) \mathrm{diff}\left(t\,t\right)\+{\mathrm{D}}_2\left(f\right)\left(t\,t^2\right) \mathrm{diff}\left(t^2\,t\right)$

$-4t^3-2t$

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

$F≔f\left(t\,t^2\right)\:$

$\mathrm{diff}\left(F\,t\right)$

$-4t^3-2t$