Suppose $f(r\mid \theta)$ is double exponential distribution with pdf as
$$f(r\midθ) = \frac{1}{2\sigma}\exp\left({−\frac{|r − \mu|}{\sigma}}\right)$$
, where $\theta= (\mu,\sigma)\,, −\infty<\mu<\infty,\,\sigma>0$.
Now given a random sample $r_1,r_2,r_3,\ldots,r_n$ find the MLE of $\theta$.
So, I can't solve this with calculus because the partial derivative of $f$ w.r.t $\mu$ doesn't exist everywhere, right? Or can I? If I can how do I justify this?