How to differentiate this exponential process?

Question

I understand how to apply Ito's formula to differentiate the Radon-Nikodym density $Z_t = \exp \bigg(-\int_0^t \theta_sdW_s - \frac{1}{2} \int_0^t \theta^2_sds \bigg)$ and get the SDE $dZ_t = -\theta_tZ_tdW_t$. However, I don't know how to prove that the derivative of $$K_t = \exp \bigg(\int_0^t \ln(1+\eta_s) dH_s - \int_0^t \eta_s \zeta_s ds \bigg)$$ is the SDE $$dK_t = \eta_tK_t (dH_t - \zeta_tdt)$$ where $H_t = I_{\{ \tau \leq t \}}$, $\tau$ is a random time, $\zeta_t$ & $\eta_t$ are a predictable processes and $dH_t-\zeta_tdt$ is a martingale.