That one is so bad, because it doesn't work. Presumably the derivative is used to calculate a tangent line. However, a tangent line touches the curve only at one point, unless the curve is totally flat. Either way, this is not what you want. You want fuckin' Taylor series approximations. (They won't work on curves that aren't smooth, but that's all right, because those are the curves you want to touch anyway.)
Strictly speaking, the Taylor series won't even do what you're wanting them to do for smooth functions. (Consider the function e^{-1/x^2}, whose Taylor series is zero at the origin. In fact, given any sequence of numbers, you can cook up a smooth function which has that sequence as the coefficients of its Taylor series at zero.) If you want the Taylor series to agree with the function in an entire neighborhood of a point, you want the function to be real (or complex) analytic (at least in a neighborhood of that point).
That's true. I was careful to say "They won't work on curves that aren't smooth", which doesn't mean "They will work on all curves that are smooth", so I think what I said is still correct.
And as long as we are going for perfect rigor, the function you describe is undefined at x=0; you have to make it piecewise and say "if x=0, then this function is 0". (And I could say something about "you want natural curves, not this artificial pieced-together stuff.")
Also, I believe the Taylor series for that function is well-defined and accurate at all points other than x=0, so you can just pick one of those points. Are there smooth functions whose Taylor series are wrong everywhere? I doubt it. And one might argue that it's pretty problematic that, e.g., with the function "0 if x≤0, e^{-1/x^2} if x>0", a Taylor series at any positive point will be wrong for all negative numbers, and a Taylor series at any nonnegative point will be wrong for all positive numbers... well, I dunno, touching half of the entire range of a curve is still a lot (an infinite amount, in fact).
One could imagine a bunch of functions pieced together to make a smooth function (e.g. "e^{-1/x^2} * e^{-1/(x-1)^2} for 0<x<1, e^{-1/(x-1)^2} * e^{-1/(x-2)^2} for 1<x<2, ...") whose Taylor series are all accurate only over a small, finite domain. I guess that kinda answers my question. But at this point I would apply the "natural curves" objection. I wonder, are there smooth non-piecewise functions whose Taylor series are all wrong like that? (Absolute value is a piecewise function, so I'd probably reject anything with absolute value.)
Helicase is what unzips the genes, not RNA polymerase. RNA polymerase allows transcription (or creation of an mRNA molecule complementary to the given gene) to take place.
I wish I was your derivative, so I could lie tangent to your curves.