The answer I give my students is that mathematicians want to know what a word (in this case 'derivative') means in all cases, and the definition of the derivative is a communal agreement about what to say in strange cases such as the absolute value function. (Well, since I banish symbolic stuff from the first two weeks, I say 'function whose graph has a sharp corner like this one (draws on board)'.)

If students press further, I point out that in a literature class they are expected to learn the communal agreement on the difference between a 'simile' and a 'metaphor'. It helps that I am at a liberal arts institution and not a technical one.

Let me also use this opportunity to share a pedagogical trick:

I find it helpful (third time I've tried it) to break up the definition of $f^\prime(2)$ into two parts:

1. Define a new function $E_2$ by the formula $$E_2(x)=\frac{f(x)-f(2)}{x-2}.$$
2. Take the limit of $E_2$ at 2.

To pull this off, you do need to take the function $E_2$ somewhat seriously; graph it, write formulas for it, et c.

Rationale:

1. It always helps to break up complicated definitions into smaller pieces.

2. It emphasizes that you take limits of functions (in the sense of machines that accept a single number as input and gives a single number of output) rather than of symbolic expressions.

3. Students get to really understand why a discontinuous function or something like the absolute value function is not differentiable (at the relevant point).