Let $W^{1,2}=\{X\in L^2([0,T]\times \Omega ), X'\in L^2([0,T]\times \Omega ) \}$.
Let $$X_t=\int_0^t f(X_s)ds+W_t,$$ where $f$ is Lipschitz continuous with linear growth and $W$ is a Brownian motion. We set $Z_t=X_t-W_t$. Then, in my exercise they claim that $X$ and $Z$ are in $W^{1,2}$. So, I agree that $Z\in W^{1,2}$ since $Z_t'=f(X_t)$. But why $X\in W^{1,2}$ ? What is the signification of $X_t'$ ?