Help in derivative with summation

Question

I have forgotten how to handle the derivative $\frac{\partial}{\partial x}[(n-1) \sum_{y_i} \log x({y_i})]$ where $x$ is a function of a vector $\mathbf{y_i}$. How do I evaluate this? Thank you

Answer 1

Since we are taking the derivative with respect to $x$ and not $\boldsymbol{y_i}$, we can essentially view $x$ as a variable for the differentiation, and then evaluate the result at the $\boldsymbol{y_i}$. That is, what $x$ does to $\boldsymbol{y_i}$ as a function does not affect the differentiation that you are interested in. So we can treat this as a relatively simple differentiation exercise:

\begin{eqnarray} \frac{\partial}{\partial x} \left[ (n-1) \sum_{y_i} \log x(\boldsymbol{y_i}) \right] &= (n-1) \sum_{y_i} \frac{\partial}{\partial x} \left[ \log x(\boldsymbol{y_i}) \right] \\ &= (n-1) \sum_{y_i} \frac{1}{x(\boldsymbol{y_i})} \end{eqnarray}

Recall that the derivative of $\log(x)$ is $1/x$.