A derivation means a series of logical steps that starts with some assumptions, and ends up at the result you want. Just about anything can be "derived", as long as you vary what the assumptions are. So when people say "X can't be derived", they mean "at your current level of understanding, there's no way to derive X that sheds more light on why X is true, over just assuming it is".
For example, can you "derive" that momentum is $p = mv$? There are several possible answers.
* You ask this as a student in introductory physics. Some might say yes. For example, you can start from the kinetic energy $K = mv^2/2$, and then assume $K = p^2/2m$. Combining these equations and solving for $p$ gives $p = mv$, so this is a derivation.
* You ask this as a student in introductory physics. Some might say no. The above derivation is just nonsense. Starting from $K = p^2/2m$ is basically the same thing as assuming the final result, and if you're allowed to do that, it's no better than just taking $p = mv$ by definition. It's like "deriving" $1 + 1 = 2$ by defining $2$ to be $1 + 1$.
* You ask this as a student in advanced mechanics. Most would say yes. You start from the deeper idea that symmetries are related to conserved quantities, along with the definition that momentum should be the conserved quantity associated with translational symmetry. 28
Putting these together gives the result.
The point is, you can make up a derivation for anything -- but you might not be at a stage in your education where such a derivation is useful at all. If the derivation only works by making up ad-hoc assumptions that are basically as unmotivated as what you're trying to prove, then it doesn't aid understanding. Some people feel this is true for the <PERSON> equation, though I personally think its elementary derivations are quite useful. (The classic one is explained in a later answer here.)
There is often confusion here because derivations in physics work very differently than proofs in mathematics.
For example, in physics, you can often run derivations in both directions: you can use X to derive Y, and also Y to derive X. That isn't circular reasoning, because the real support for X (or Y) isn't that it can be derived from Y (or X), but that it is supported by some experimental data D. This two-way derivation then tells you that if you have data D supporting X (or Y), then it also supports Y (or X).
Once you finish putting high school math on a rigorous foundation, undergraduate math generally builds upward. For example, you can't use <PERSON>' theorem to prove the fundamental theorem of calculus, even though it technically subsumes it as a special case, because its proof depends on the fundamental theorem of calculus in the first place. In other words, as long as your classes are being rigorous at all, it would be very strange to hear "we can't derive this important result now, but we'll derive it next year" -- that would be in danger of logical circularity.
This isn't the case in physics: undergraduate physics generally builds downward. Every year, you learn a new theory that subsumes everything you previously learned as a special case, which is completely logically independent of those earlier theories. You don't actually need any results from classical mechanics to completely define quantum mechanics: it is a new layer constructed below classical mechanics rather than above it. That's why definitions now can turn into derived things later, once you learn the lower level. And it means that in practice, physicists have to guess the lower level given only access to the higher level; that's the fundamental reason why science is hard! 855