Find the maximum likelihood estimator for $\theta$ when $f(x)=2\theta^{-2}x, 0\leq x \leq \theta$

There's an issue that arises here. Your distribution is supported (nonzero) for $0\leq x\leq\theta$. In other words, it's zero outside this interval. That means when you look at something like $f(x_i)=2\theta^{-2}x_i$, it should actually be $f(x_i)=2\theta^{-2}x_i\mathbb{1}_{0\leq x_i\leq \theta}$, where the indicator is zero when $x_i$ is outside the interval. When you are maximizing $f(x_1)\cdots f(x_n)$, this product is zero if $\theta<x_i$ for some $i$, so you can just assume that $\theta\geq x_i$ for each $i$.

So think about how you would maximize such a function. Don't deal with derivatives because the function won't have a zero derivative, but will have a maximum. As a hint, try looking at $\hat{\theta}=\max_i x_i$. Can you do better than that?