I'll use subscript notation for brevity, i.e.
[tex]\dfrac{\partial f}{\partial x}=f_x[/tex]
3.
[tex]f(x,y,z)=xy+z^2\implies\begin{cases}f_x=y\\f_y=x\\f_z=2z\end{cases}[/tex]
[tex]x(r,s)=s^2\implies\begin{cases}x_r=0\\x_s=2s\end{cases}[/tex]
[tex]y(r,s)=2rs\implies\begin{cases}y_r=2s\\y_s=2r\end{cases}[/tex]
[tex]z(r,s)=r^2\implies\begin{cases}z_r=2r\\z_s=0\end{cases}[/tex]
By the chain rule,
[tex]f_r=f_xx_r+f_yy_r+f_zz_r=2xs+4zr=2s^3+4r^3[/tex]
[tex]f_s=f_xx_s+f_yy_s+f_zz_s=2ys+2xr=6rs^2[/tex]
4.
[tex]x(r,s,t)=r+s-2t\implies\begin{cases}x_r=1\\x_s=1\\x_t=-2\end{cases}[/tex]
[tex]y(r,s,t)=3rt\implies\begin{cases}y_r=3t\\y_s=0\\y_t=3r\end{cases}[/tex]
[tex]z(r,s,t)=s^2\implies\begin{cases}z_r=0\\z_s=2s\\z_t=0\end{cases}[/tex]
By the chain rule,
[tex]f_r=f_xx_r+f_yy_r+f_zz_r=y+3xt=3rt+3r+3s-6t^2[/tex]
[tex]f_s=f_xx_s+f_yy_s+f_zz_s=y+4zs=3rt+4s^3[/tex]
[tex]f_t=f_xx_t+f_yy_t+f_zz_t=-2y+3xr=3r+3s-12rt[/tex]
