Find the MLE of the unknown parameter $\displaystyle \theta$ when $\displaystyle X_{1},...X_{n}$ is a sample from the following distribution:

$\displaystyle f_{X}(x) = \frac{1}{2}e^{-|x-\theta|}, -\infty < x < \infty$

I have the following, not sure if it's correct.

$\displaystyle f(x_{1}...x_{n}|\theta) = \prod_{i=1}^{n} \frac{1}{2}e^{-|x_{i}-\theta|}$

$\displaystyle =\frac{1}{2^{n}}e^{-[\sum_{i=1}^{n}x_{i}] - n\theta}$

$\displaystyle \log f(x_{1}...x_{n}|\theta) = \frac{1}{2^{n}} [-\sum_{i=1}^{n}x_{i} - n\theta]$

$\displaystyle \frac{d}{d\theta}\log f(x_{1}...x_{n}|\theta) = \frac{-n}{2^{n}}$

Is this correct?