I studied this 10+ years ago. But I just saw a very concise derivation from stack exchange

Let $N_t$ be a Poisson counter process, with parameter $\lambda>0$. Let there be another stochastic process $X_t = f(t,N_t)$. Then, by Taylor series expansion,

The second equality is due to the fact that $dN_tdt = 0$, and $dN_t = (dN_t)^2 = (dN_t)^3 = \cdots = (dN_t)^n$ since $dN_t \in \{0, 1\}$.

Therefore, we have

For example, if

then