(Hooke's Law for Biomolecules) The entropy of conformation of a protein, considering it to be a freely jointed chain that contains a total of N segments, each of Length L, is given by the expression: $ S = \frac{1}{2} NK_b [ (1 + \alpha) \ln(1 + \alpha) + (1 - \alpha) \ln(1 - \alpha)$ where $\alpha = x/NL$ is the fractional extension and $K_b$ is the boltzmann constant. At constant volume, the extensional force is given by $F=T \frac{ds}{dx}$. For a small extension $(\alpha \ll 1)$, show that the Hooke's Law is obeyed. (Hooke's law states that $F$ is proportional to $x$)
The words "small extension" screams to me that this is an application of the power series expansion question.
First I am trying to re-express $F=T \frac{ds}{dx}$ but the function $S$ does not contain $x$, how do I go about expressing this?