What are the slopes of vertical lines?

Answer 1

Undefined.

Proof, let a function f have a vertical line on x = c.

(Notice: By definition of functions, it is not even a function, which means that we do not even need to discuss differentiability. We assume it is a function)

Now suppose f'(c) exist, then

f'(c) = lim x --> c (f(x) - f(c))/(x - c), the limit exist but since it's a straight line, assume non trivial (a point), we have automatically x = c. But since it's non-trivial, hence f(x) != f(c), let f(x) - f(c) = r for some real number r != 0.

we get f'(c) = lim x --> c (f(x) - f(c)) / (x - c) = r/0 which is undefined.

Contradiction!.

Hence f'(c) doesn't exist.

Note: If you see a straight line some where on a "function" and they ask for derivative, write:"This is not a function, what kind of question is this! Go back to Calculus class!"

If you are discussing a function with a vertical slope, e.g. let f(x) = cubeRoot(x), then it's a different proof.