Differentiable

When we say that a function is differentiable in an interval, it means that this function’s derivative exists at each point in this interval.

When we say derivative exists at certain point a, it means we can pick a new point a+c and use function at a+c minus function at a, then divided by c.

When a+c is approaching a, which means c is sufficiently small, if the limit of above equation still exists, then we can say it is derivative at this point.

If a function can be derivative at every point in a interval, we can say this function is differentiable in this interval.