$F_r$

$\=f_x x_r\+f_y y_r$

$F_t$

$\=f_x x_t\+f_y y_t$

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

$\=-3x^{2}yr\sin \left(t\right)\+x^{3}r\cos \left(t\right)$

$\=4r^3{\cos \left(t\right)}^3\sin \left(t\right)$

$\=-3r^4{\cos \left(t\right)}^2{\sin \left(t\right)}^2\+r^4{\cos \left(t\right)}^4$

$\=r^4\left(4{\cos \left(t\right)}^2-3\right)\cos {\left(t\right)}^2$

Formal statement of the relevant chain rules

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

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

Implement the chain rule

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

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

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

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

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

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

$4r^3{\cos \left(t\right)}^3\sin \left(t\right)$

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

$-3x^{2}yr\sin \left(t\right)\+x^{3}r\cos \left(t\right)$

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

$-3r^4{\cos \left(t\right)}^2{\sin \left(t\right)}^2\+r^4{\cos \left(t\right)}^4$

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

$\left(4{\cos \left(t\right)}^2-3\right){\cos \left(t\right)}^2{r}^4$

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

$\frac{\partial }{\partial r} f\left(X\,Y\right)$ = $4r^3{\cos \left(t\right)}^3\sin \left(t\right)$$$$$

$\frac{\partial }{\partial t} f\left(X\,Y\right)$ = $-3r^4{\cos \left(t\right)}^2{\sin \left(t\right)}^2\+r^4{\cos \left(t\right)}^4$$\stackrel{\text{simplify}}{\=}$$\left(4{\cos \left(t\right)}^2-3\right){\cos \left(t\right)}^2{r}^4$$$$$

Initialize

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

Formal statement of the relevant chain rules

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

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

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

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

Implement the chain rule

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

$f≔\left(x\,y\right)\to x^{3}y\:$

$$\begin{gathered} X≔r \cos \left(t\right)\: \\ Y≔r \sin \left(t\right)\: \end{gathered}$$$$

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

$4r^3{\cos \left(t\right)}^3\sin \left(t\right)$

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

$\left(4{\cos \left(t\right)}^2-3\right){\cos \left(t\right)}^2{r}^4$

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

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

$4r^3{\cos \left(t\right)}^3\sin \left(t\right)$

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

$\left(4{\cos \left(t\right)}^2-3\right){\cos \left(t\right)}^2{r}^4$