Question: I am solving a problem and just can't wrap my had around the result I'm getting...
Here it is: ![enter image description here](https://i.stack.imgur.com/5XGaH.jpg)

So my next step would be setting derivative equal to zero and solving for Theta...
Are my calculations wrong?...
I would appreciate any constructive advice.

Answer: 
You are getting stuck because you are treating $L$ as if it was differentiable everywhere. Note that if $\theta > x\_k$ for any $k$, then $L(\theta) = 0$. If $\theta \le \min\_k x\_k$, then $L$ is an increasing function of $\theta$. Hence it is maximised when $\theta = \min\_k x\_k$.
So, thanks to the great advices from @copper.hat and @AlexR I made an attempt to explain the solution using the definition of MLF:
The function in 1) is an increasing function of the argument Theta, thus it reaches its maximum value at maximum value of its argument. Since we have the restriction on Theta, namely:Theta is less or equal to x, the maximum possible value of Theta will be: minimal of (X1,...,Xn),
Thus the MLE of Theta is min(X1,...,Xn)

Need critique on that.