derivative
Picture driving a car and glancing at the speedometer. The odometer tells you how far you have gone (a function of time), but the speedometer tells you how fast you are going right now. That instantaneous speed is a derivative: the rate at which one quantity changes with respect to another at a single instant, not averaged over an interval.
Precisely, the derivative of a function f at a point a is the limit f'(a) = lim_{h -> 0} (f(a + h) - f(a)) / h, provided this limit exists and is a finite real number. The quotient inside is a difference quotient, and the derivative is what it approaches as the input increment h shrinks toward zero. Equivalently, writing x = a + h, f'(a) = lim_{x -> a} (f(x) - f(a)) / (x - a).
The honest subtlety is that this is a limit, and limits can fail to exist. At a corner like f(x) = |x| at 0 the left and right difference quotients tend to different values (-1 and +1), so there is no single derivative. Existence of the derivative is therefore a genuine hypothesis, not something to assume; a function can be perfectly continuous yet have no derivative at a point, or even at every point.
For f(x) = x^2 and a = 3, the difference quotient is ((3 + h)^2 - 9) / h = (6h + h^2) / h = 6 + h for h not 0, which tends to 6 as h -> 0. So f'(3) = 6.
The algebra cancels the h in the denominator before the limit is taken — exactly why the 0/0 form is not a problem.
Notation varies: f'(a), df/dx at a, and Df(a) all mean the same number. The Leibniz symbol df/dx is suggestive but is not literally a fraction; it is a single symbol for the limit.