Fair enough, I agree that manifolds are a poor example. Let me try another one, which I hope will illustrate my point more effectively. Consider the article continuous function. This is certainly something that a first year college student might look up (possibly even a high school student). I think continuous function does a much better job of handling different audiences than the current version of derivative (although incidentally in my opinion it still needs a lot of work... some pictures would be nice). It starts out with a fairly wishy-washy description in terms of input values having effects on output values, without saying too much about what kind of functions are being discussed. Then it gives some specific examples, which happen to be real-valued functions of real numbers. Then it has a paragraph discussing generalisations, which link immediately to Continuity (topology) which gives the most general definitions of continuity in mathematical use. The next section of the article is called "Real-valued continuous functions" and is very explicit about its restricted scope. I think that the Continuous function article manages all of this without in any way "scaring off" a first-year undergrad; if anything it might just pique their curiosity. 

 In contrast, the introduction to Derivative gives not the slightest indication that the concept leads to something far more general. There's a really nice sentence later in Derivative, The common thread is that the derivative at a point serves as a linear approximation of the function at that point. I think it would be great if something like that could appear in the intro. I'm not going to write it now because there seems to be a lot of opposition, but if I get any support here (HINT HINT), maybe I'll try.