Evaluating partial derivative of a double summation

Question

Suppose I am given a function $f(x_i,y_j) = \sum_{i=1}^n\sum_{j=1}^m y_j\log(x_i)$.

I want to evaluate $\frac{\partial f}{\partial x_i}$. Am I correct in assuming that

$\frac{\partial f}{\partial x_i} = \frac{\sum_{j=1}^m y_j}{x_i}$, as the partial derivative of all $x_{i'}, i' \neq i$ equals 0, so the summation drops out?

Answer 1

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

$$\frac{\partial f}{\partial x_i}=\frac{1}{x_i}\Sigma_{j=1}^m y_j$$

$i=1,2,\dots,n$