= Solution
Take outward radial <force> as positive and keep the displaced element's angular rate fixed at $\Omega_0=\Omega(R_0)$. Its imbalance is
$$
\Delta f_R=\rho\left[(R_0+\xi_R)\Omega_0^2-\Phi'(R_0+\xi_R)\right].
$$
Use the equilibrium relation from part <1/b>, $\Phi'(R)=R\Omega(R)^2$, and a first-order <Taylor expansion>:
$$
\boxed{\Delta f_R=-\rho R_0\left.\frac{d\Omega^2}{dR}\right|_{R_0}\xi_R+O(\xi_R^2),\qquad
|\Delta f_R|\simeq\rho\left|R_0\frac{d\Omega^2}{dR}\xi_R\right|.}
$$
The absolute values are needed for a magnitude. As printed, the proposed magnitude has a signed right side; it would be negative for outward displacement in a <Keplerian disk>. For decreasing $\Omega^2$, the signed imbalance above points outward and therefore amplifies that fixed-angular-rate displacement. This calculation fixes the angular rate, rather than the <specific angular momentum>; the latter is the conserved quantity for an uncoupled inviscid radial displacement and leads to the <Rayleigh centrifugal stability criterion>.
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