Assign the identity $f\to F$ the name $q$

$f\left(r \cos \left(t\right)\,r \sin \left(t\right)\right)\=F\left(r\,t\right)$$\stackrel{\text{assign to a name}}{\to }$$q$

Differentiate and solve for $f_x$ and $f_y$ 

$\frac{\partial }{\partial r} q\,\frac{\partial }{\partial t} q$

${\mathrm{D}}_1\left(f\right)\left(r\cos \left(t\right)\,r\sin \left(t\right)\right)\cos \left(t\right)\+{\mathrm{D}}_2\left(f\right)\left(r\cos \left(t\right)\,r\sin \left(t\right)\right)\sin \left(t\right)\=\frac{\partial }{\partial r}F\left(r\,t\right)\,-{\mathrm{D}}_1\left(f\right)\left(r\cos \left(t\right)\,r\sin \left(t\right)\right)r\sin \left(t\right)\+{\mathrm{D}}_2\left(f\right)\left(r\cos \left(t\right)\,r\sin \left(t\right)\right)r\cos \left(t\right)\=\frac{\partial }{\partial t}F\left(r\,t\right)$

$\stackrel{\text{solve (specified)}}{\to }$

$\left\{{\mathrm{D}}_1\left(f\right)\left(r\cos \left(t\right)\,r\sin \left(t\right)\right)\=-\frac{-\cos \left(t\right)\left(\frac{\partial }{\partial r}F\left(r\,t\right)\right)r\+\left(\frac{\partial }{\partial t}F\left(r\,t\right)\right)\sin \left(t\right)}{r\left({\cos \left(t\right)}^2\+{\sin \left(t\right)}^2\right)}\,{\mathrm{D}}_2\left(f\right)\left(r\cos \left(t\right)\,r\sin \left(t\right)\right)\=\frac{\left(\frac{\partial }{\partial r}F\left(r\,t\right)\right)r\sin \left(t\right)\+\left(\frac{\partial }{\partial t}F\left(r\,t\right)\right)\cos \left(t\right)}{r\left({\cos \left(t\right)}^2\+{\sin \left(t\right)}^2\right)}\right\}$

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

$\left\{{\mathrm{D}}_1\left(f\right)\left(r\cos \left(t\right)\,r\sin \left(t\right)\right)\=\frac{\cos \left(t\right)\left(\frac{\partial }{\partial r}F\left(r\,t\right)\right)r-\left(\frac{\partial }{\partial t}F\left(r\,t\right)\right)\sin \left(t\right)}{r}\,{\mathrm{D}}_2\left(f\right)\left(r\cos \left(t\right)\,r\sin \left(t\right)\right)\=\frac{\left(\frac{\partial }{\partial r}F\left(r\,t\right)\right)r\sin \left(t\right)\+\left(\frac{\partial }{\partial t}F\left(r\,t\right)\right)\cos \left(t\right)}{r}\right\}$

Notational simplifications

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

Differentiate the equation $f\left(x\,y\right)\=F\left(r\,t\right)$ with respect to $r$ and $t$ 

$$\begin{gathered} q_r≔\mathrm{diff}\left(f\left(x\,y\right)\=F\,r\right)\; \\ {q}_t≔\mathrm{diff}\left(f\left(x\,y\right)\=F\,t\right) \end{gathered}$$

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

Solve the two equations for $f_x$ and $f_y$ 

$q_1≔\mathrm{solve}\left(\left\{q_r\,q_t\right\}\,\left\{{\mathrm{D}}_1\left(f\right)\left(x\,y\right)\,{\mathrm{D}}_2\left(f\right)\left(x\,y\right)\right\}\right)\:$

Display the two solutions on separate lines

$\mathrm{print}~\left(q_1\right)\left[\right]$

${\mathrm{D}}_1\left(f\right)\left(x\,y\right)=-\frac{\left(y_r\right)\left(F_t\right)-\left(y_t\right)\left(F_r\right)}{\left(y_t\right)\left(x_r\right)-\left(y_r\right)\left(x_t\right)}$

Replace the names $x$ and $y$ with $r \cos \left(t\right)$ and $r \sin \left(t\right)$, respectively; display the results

$q_2≔\mathrm{expand}\left(\mathrm{simplify}\left(\mathrm{eval}\left({q_1}_{}\,\left[x\=r \cos \left(t\right)\,y\=r \sin \left(t\right)\right]\right)\right)\right)\:$

$\mathrm{print}~\left(q_2\right)\left[\right]$

${\mathrm{D}}_1\left(f\right)\left(r\cos \left(t\right)\,r\sin \left(t\right)\right)=-\frac{\sin \left(t\right)\left(F_t\right)}{r}+\cos \left(t\right)\left(F_r\right)$