Non-infinitesimal linear expansion of a differential operator $X$

Question

Consider a differential equation of the form $$ \frac{df}{dt} = X(f) $$ where $X$ is a linear operator. Then, if I am not mistaken, the infinitesimal change of $f$ from time $t$ to $dt$ is given as: $$ f(t+dt) = e^{dt X}f $$ which can be expanded as $$ f(t+dt) = (1+Xdt)f + \text{ higher order terms}. $$ My question is how can I compute a similar expansion for a non-infinitesimal but arbitrarily small change $t \to t + \Delta t$ and whether it still holds that to first order $$ f(t+\delta t) = (1+X \delta t)f + \text{ higher order terms}. $$ If the equation above is not correct, what is the best way to linearly approximate $e^{\delta t X}$ acting on $f$?