Answer
$$\frac{19}{15}$$
Work Step by Step
Given $$f(x, y)=\ln (x y), \ \ \ x=3 r+2 s,\ \ \ y= 5r+3s ,\ \ \ (r,s)=(1,0)$$
Since at $(r,s)=(1,0)$; $(x,y)=( 3,5)$ and
\begin{align*}
\frac{\partial f}{\partial x}&= \frac{1}{x},\ \ \ \ \ \ \frac{\partial f}{\partial y} = \frac{1}{y}\\
\frac{\partial x}{\partial r}&= 3,\ \ \ \ \ \ \frac{\partial x}{\partial s} = 2\\
\frac{\partial y}{\partial r}&= 5,\ \ \ \ \ \ \frac{\partial y}{\partial s} = 3\\
\end{align*}
Then
\begin{align*}
\frac{\partial f}{\partial s}&=\frac{\partial f}{\partial x}\frac{\partial x}{\partial s}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial s}\\
&=\frac{2}{x} +\frac{3}{y}\\
\frac{\partial f}{\partial s}\bigg|_{(3,5)}&= \frac{2}{3} +\frac{3}{5}=\frac{19}{15}
\end{align*}
