I've trying to get the maximum likelihood estimator of $\theta_{MLE}$ but after doing the final derivate step. I've got -1. Am I doing the partial derivative wrongly? What is the MLE of $\theta$?
$$ \begin{equation} f\left(x|\theta\right)= \begin{cases} e^{-(x-\theta)} & x\geq\theta \\ 0 & \text{otherwise.} \end{cases} \end{equation} $$
Negative log-likelihood \begin{split} L(x|\theta) &= -log(f(x|\theta))\\ &= \sum_{i=1}^n (x_i-\theta) \end{split}
Partial derivative \begin{split} \frac{\partial L(x|\theta)}{\partial\theta} &= \frac{\partial}{\partial\theta}\left(\sum_{i=1}^n (x_i-\theta)\right) &= -1 \end{split}
The negative log likelihood is $\sum_{i=1}^n (x_i - \theta)$ when $\theta \le \min_i x_i$, and is $\infty$ otherwise. To minimize the negative log likelihood, you should therefore choose $\theta = \min_i x_i$. No need for partial derivatives.