= Differential magnetic susceptibility
{title2=$\chi_{\rm diff}=\left(\partial M/\partial H_{\rm int}\right)_T$}
<Differential magnetic susceptibility> is the local slope at a specified bias <magnetic field intensity>, holding <temperature> and other stated constraints fixed. It equals <volume magnetic susceptibility> only for a linear response through the origin. For example, $M=M_s\tanh(H/H_s)$ has $M/H=(M_s/H_s)\tanh(x)/x$, but $dM/dH=(M_s/H_s)/\cosh^2(x)$, where $x=H/H_s$. At $x=1$ these are about $0.762$ and $0.420$ times their common zero-field slope. A finite sweep chord is another quantity, not automatically this <derivative>.
If the differentiated field is instead the internal <magnetic flux density>, the longitudinal <chain rule> applied to $B_{\rm int}=\mu_0(H_{\rm int}+M)$ gives
$$
\left(\frac{\partial M}{\partial B_{\rm int}}\right)_T
=\frac{\chi_{\rm diff}}{\mu_0(1+\chi_{\rm diff})}.
$$
This differs from the vacuum/applied-$B_0$ conversion in <volume magnetic susceptibility>. It assumes a differentiable single branch with locally invertible $B_{\rm int}(H_{\rm int})$.
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