$F_r$

$\=f_x x_r\+f_y y_r$

$\=\frac{18x}{3x^2\+4y^2}\+\frac{40y}{3x^2\+4y^2}$

$\=\frac{18\left(3r\+2s\right)}{3{\left(3r\+2s\right)}^2\+4{\left(5r-7s\right)}^2}\+\frac{40\left(5r-7s\right)}{3{\left(3r\+2s\right)}^2\+4{\left(5r-7s\right)}^2}$

$\=\frac{2\left(127r-122s\right)}{127r^2-244rs\+208s^2}$

$F_s$

$\=f_x x_s\+f_y y_s$

$\=\frac{12x}{3x^2\+4y^2}-\frac{56y}{3x^2\+4y^2}$

$\=\frac{12\left(3r\+2s\right)}{3{\left(3r\+2s\right)}^2\+4{\left(5r-7s\right)}^2}-\frac{56\left(5r-7s\right)}{3{\left(3r\+2s\right)}^2\+4{\left(5r-7s\right)}^2}$

$\=\frac{4\left(104 s-61 r\right)}{127r^2-244rs\+208s^2}$

Formal statement of the relevant chain rules

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

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

Implement the chain rule

$f\left(x\,y\right)\=\ln \left(3x^2\+4y^2\right)$$\stackrel{\text{assign as function}}{\to }$$f$$$

$X\=3 r\+2 s$$\stackrel{\text{assign}}{\to }$$$

$Y\=5 r-7 s$$\stackrel{\text{assign}}{\to }$$$

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

$\frac{18x}{3x^2\+4y^2}\+\frac{40y}{3x^2\+4y^2}$

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

$\frac{18\left(3r\+2s\right)}{3{\left(3r\+2s\right)}^2\+4{\left(5r-7s\right)}^2}\+\frac{40\left(5r-7s\right)}{3{\left(3r\+2s\right)}^2\+4{\left(5r-7s\right)}^2}$

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

$\frac{2\left(127r-122s\right)}{127r^2-244rs\+208s^2}$

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

$\frac{12x}{3x^2\+4y^2}-\frac{56y}{3x^2\+4y^2}$

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

$\frac{12\left(3r\+2s\right)}{3{\left(3r\+2s\right)}^2\+4{\left(5r-7s\right)}^2}-\frac{56\left(5r-7s\right)}{3{\left(3r\+2s\right)}^2\+4{\left(5r-7s\right)}^2}$

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

$-\frac{4\left(61r-104s\right)}{127r^2-244rs\+208s^2}$

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

$\frac{\partial }{\partial r} f\left(X\,Y\right)$ = $\frac{254r-244s}{3{\left(3r\+2s\right)}^2\+4{\left(5r-7s\right)}^2}$$\stackrel{\text{simplify}}{\=}$$\frac{2\left(127r-122s\right)}{127r^2-244rs\+208s^2}$ $$$$

$\frac{\partial }{\partial s} f\left(X\,Y\right)$ = $\frac{-244r\+416s}{3{\left(3r\+2s\right)}^2\+4{\left(5r-7s\right)}^2}$$\stackrel{\text{simplify}}{\=}$$-\frac{4\left(61r-104s\right)}{127r^2-244rs\+208s^2}$$$$$

Initialize

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

Formal statement of the relevant chain rules

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

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

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

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

Implement the chain rule

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

$f≔\left(x\,y\right)\to \ln \left(3x^2\+4y^2\right)\:$

$$\begin{gathered} X≔3 r\+2 s\: \\ Y≔5 r-7 s\: \end{gathered}$$$$

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

$\frac{2\left(127r-122s\right)}{127r^2-244rs+208s^2}$

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

$-\frac{4\left(61r-104s\right)}{127r^2-244rs+208s^2}$

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

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

$\frac{2\left(127r-122s\right)}{127r^2-244rs+208s^2}$

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

$-\frac{4\left(61r-104s\right)}{127r^2-244rs+208s^2}$