Ramification divisors for normal varieties

Question

Let $X$ be a normal variety and $Y$ be smooth variety with a dominant, generically finite, separable morphism, $$ \pi: Y \longrightarrow X. $$ It seems to be folklore that there is an effective Cartier divisor $R$ on $Y$ such that $$ K_Y \sim \pi^* K_X + R. $$ Can someone please explain in some detail why this is true? In the case that both $X$ and $Y$ are smooth, then I believe we are able to obtain an inclusion $\pi^* K_X \subseteq K_Y $ and proceed from there. But that doesn't work when $X$ is not smooth so the sheaf of differentials is no longer locally free. Is something able to explain how this ramification formula arises?