Find the MLE of the distribution having the following pdf

$\displaystyle f(x;\theta)=e^-(x-\theta)$, $\displaystyle \theta \le x<\infty$, $\displaystyle -\infty<\theta<\infty$

Attempt to a solution:

Likelihood function

$\displaystyle L(\theta;X_i)=e^[-\sum X_i+\theta]$

Log likelihood function

$\displaystyle l(\theta)=-\sum X_i+\theta$

But then taking the derivative with respect to $\displaystyle \theta$ it makes $\displaystyle \theta$ disappear.

I feel like i missing something especially with the bounds of x and $\displaystyle \theta$