Wait for the right moment

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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 $f'(3) = \lim _{h\to 0}\frac {f(3+h)-f(3)}{h}.$ So I guess before.
Devyn: Aha! I think you’re wasting time. You see I write $f'(x) = \lim _{h\to 0}\frac {f(x+h)-f(x)}{h}.$ 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. $g'(2) = \answer {4}$