Just to visualize the problem (I think it can be useful) consider that your double sum is the sum of the elements of the following matrix
$$ \left[ \begin{matrix}
y_1\log x_1& y_1\log x_2 & \dots & y_1\log x_n \\
y_2\log x_1& y_2\log x_2 & \dots & y_2\log x_n \\
\dots & \dots & \dots &\dots \\
y_m\log x_1 & y_m\log x_2 & \dots & y_m\log x_n \\
\end{matrix}\right]$$
thus derivating w.r.t. $x_i$ the result is the sum of the elements of the following matrix
$$ \left[ \begin{matrix}
y_1/x_1& y_1/x_2 & \dots & y_1/x_n \\
y_2/x_1& y_2/x_2 & \dots & y_2/ x_n \\
\dots & \dots & \dots &\dots \\
y_m/x_1 & y_m/ x_2 & \dots & y_m/x_n \\
\end{matrix}\right]$$
That is
$$\frac{\partial f}{\partial x_i}=\frac{1}{x_i}\Sigma_{j=1}^m y_j$$
$i=1,2,\dots,n$