Horizon-exit logarithmic derivative in slow roll ($\frac{d}{d\log k}=\frac{1}{1-{ϵ}_H}\frac{\dot{\varphi }}{H}\frac{d}{d\varphi }\simeq -\frac{V^{\prime }}{V}\frac{d}{d\varphi }$)

Along cosmological horizon exit $k\propto aH$, differentiation gives $d\log k/d\log a=1-{ϵ}_H$, where ${ϵ}_H=-\dot{H}/H^2$. The chain rule gives the first expression. In reduced Planck units, the slow-roll equations $3H\dot{\varphi }\simeq -V^{\prime }$ and $3H^2\simeq V$ give the second expression to leading order. Applied to a quantity already of slow-roll order, the omitted factor changes the answer only at second order.