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?