= Solution
Put $S(P)=[s]/c_v$. Differentiating the expression in part (c) gives
$$
S'(P)=\frac{(\gamma^2-1)(P-1)^2}{P[(\gamma+1)P+\gamma-1][(\gamma-1)P+\gamma+1]}.
$$
For $P=1+\delta$, the denominator is $4\gamma^2+O(\delta)$, so
$$
S'(1+\delta)=\frac{\gamma^2-1}{4\gamma^2}\delta^2+O(\delta^3).
$$
Integrating from the unshocked state, where $S(1)=0$, proves the cubic <entropy production in a perfect-gas shock>:
$$
\boxed{\frac{[s]}{c_v}=\frac{\gamma^2-1}{12\gamma^2}\delta^3+O(\delta^4).}
$$
The first-order and second-order terms vanish: a <weak shock> agrees with reversible compression through second order.
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