We have the following equations of state for the total magnetization \(M\), and the entropy \(S\) of a paramagnetic system:
\begin{align}
M&=N\mu\, \frac{e^{\frac{\mu B}{k_B T}} - e^{-\frac{\mu B}{k_B T}}}
{e^{\frac{\mu B}{k_B T}} + e^{-\frac{\mu B}{k_B T}}}\\
S&=Nk_B\left\{\ln 2 +
\ln \left(e^{\frac{\mu B}{k_B T}}+e^{-\frac{\mu B}{k_B T}}\right)
+\frac{\mu B}{k_B T}
\frac{e^{\frac{\mu B}{k_B T}} - e^{-\frac{\mu B}{k_B T}}}
{e^{\frac{\mu B}{k_B T}} + e^{-\frac{\mu B}{k_B T}}}
\right\}
\end{align}

List variables in their proper positions in the middle columns of the charts below.

Solve for the magnetic susceptibility, which is defined as:
\[\chi_B=\left(\frac{\partial M}{\partial B}\right)_T
\]

Using both the differentials (zapping with d) and chain rule diagram methods, find a chain rule for:

\[\left(\frac{\partial M}{\partial B}\right)_S
\]

Evaluate your chain rule. Sense-making: Why does this come out to zero?