Two young mathematicians discuss derivatives as functions.

- Devyn
- Riley, I might be a calculus genius.
- Riley
- Yeah? Explain this one to me.
- Devyn
- Let me first ask you a question. Say you have a function, like , and you want to know . Do you plug in the number before or after you find the derivative?
- Riley
- Hmmmm. Well, my next step is usually So I guess before.
- Devyn
- Aha! I think you’re wasting time. You see I write and it means that I
can look at the derivative of my function at
*any*point. So, I plug in the after I’ve found the derivative. - Riley
- That does seem like a pretty genius move. But doesn’t working with , instead of numbers, make all of this more difficult?
- Devyn
- Not at all. Let’s do the problems both ways, at the same time:
- Riley
- Whoa. So now the derivative is a function. Wait, what’s its domain? Its range?