I am having trouble understanding the following derivation.
Specifically, I am not able to understand why the partial derivative of $x_i$ with respect to $s_k$ is what it is. My hunch is that it is has something to do with the rule $ log(\frac{a}{b}) = log(a) - log(b)$. The reason being that if I am subtracting a squared value(which I generally would have expected to seen in the denominator), then it probably has something to do with logarithms. However, I am not able to derive the answer in any way.
If you apply the rule of derivation for a quotient : $ \left( \frac{u}{v} \right)' = \frac{u'v-uv'}{v^2} $, on $x_i$ it gives :
$$ \frac{\partial x_i}{\partial s_k}=\frac{\delta_{ik}e^{s_k}\left(\sum_c e^{s_c}\right)-e^{s_i}e^{s_k}}{\left(\sum_c e^{s_c}\right)^2} $$ where $\delta_{ik}=1$ if $i=k$ and $0$ if $i\neq k$.
This is exactly what you want.