= Solution
Apply the horizon-exit derivative from part (d) to the <potential slow-roll parameter> and the <second potential slow-roll parameter>. The <potential slow-roll flow equations> are
$$
\frac{d\epsilon_V}{d\log k}\simeq4\epsilon_V^2-2\epsilon_V\eta_V,
\qquad\frac{d\eta_V}{d\log k}\simeq2\epsilon_V\eta_V-\xi_V^2,
$$
where the <third potential slow-roll parameter> is $\xi_V^2=M_{\mathrm{Pl}}^4V'V'''/V^2$. For example, differentiating $\eta_V$ gives $-M_{\mathrm{Pl}}^4(V'/V)(V'''/V-V''V'/V^2)$, which is the second expression. Differentiating $n_s-1\simeq-6\epsilon_V+2\eta_V$ now gives the <running of the scalar spectral index>
$$
\boxed{\frac{dn_s}{d\log k}\simeq-6(4\epsilon_V^2-2\epsilon_V\eta_V)
+2(2\epsilon_V\eta_V-\xi_V^2)
=16\epsilon_V\eta_V-24\epsilon_V^2-2\xi_V^2.}
$$
This is the leading second-order slow-roll running. The notation $\xi_V^2$ denotes a signed derivative combination and does not force its value to be nonnegative.
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