
Two young mathematicians discuss derivatives as functions.

Check out this dialogue between two calculus students (based on a true story):
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 $f(x) = x^2$, and you want to know $f'(3)$. Do you plug in the number $3$ 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 $3$ after I’ve found the derivative.
Riley
That does seem like a pretty genius move. But doesn’t working with $x$, 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?
Suppose you have a function $f$. Which of the following are true?
The domain of $f'$ is equal to the domain of $f$. The range of $f'$ is equal to the range of $f$. The domain of $f'$ is a subset of the domain of $f$. The range of $f'$ is a subset of the range of $f$.
Find $g'(2)$ for $g(x) = x^2 + 1$ using both methods described above.