Difference Quotient and h-Method

The difference quotient (f(x₀+h) − f(x₀))/h is ordinary fraction arithmetic: the slope of the secant through two curve points, i.e. the average rate of change over an interval. The differential quotient is its limit for h → 0: the two points move together, the secant tilts into the tangent, and out comes the local rate of change f′(x₀) at a single point. Mnemonic: difference = two points, differential = one point with a limit. Example f(x) = x² at x₀ = 3: the difference quotient is 6 + h (still depends on h), the differential quotient is the number 6. Without the limit there is no derivative.

Four steps using f(x) = x², x₀ = 3. First set up: (f(3+h) − f(3))/h = ((3+h)² − 9)/h. Second expand: (9 + 6h + h² − 9)/h = (6h + h²)/h. Third factor out h and cancel: h·(6 + h)/h = 6 + h; cancelling is allowed because h ≠ 0. Fourth take the limit: for h → 0, f′(3) = 6 remains. The critical point is step 2: only when every summand in the numerator contains an h can you cancel; if an h-free term remains, there is an arithmetic error in the binomial expansion. The method works the same for x³ (with (a+b)³) or 1/x (with a common denominator).

Substituting h = 0 directly gives (f(x₀) − f(x₀))/0 = 0/0, an indeterminate expression; division by 0 is undefined. Moreover the difference quotient describes a secant through two points, and at h = 0 both would coincide: no unique line passes through a single point. The h-method avoids this cleanly: for h ≠ 0 the fraction is simplified algebraically until h is cancelled; the resulting term (say 6 + h) can then be evaluated at h = 0 too, and the limit exists. Exactly this two-step (cancel first, then h → 0) is the mathematical substance of the limit concept behind the derivative.

The average rate of change on an interval [a; b] is the difference quotient m = (f(b) − f(a))/(b − a): total change divided by interval length, geometrically the secant slope. Example: f(x) = x² on [1; 3]: m = (9 − 1)/2 = 4. In applied contexts it carries a unit, e.g. metres per second (average speed) or degrees per hour. The local rate of change f′(x₀), by contrast, is the limiting case at one point, in the speedometer picture: average speed of the trip versus instantaneous speed at the moment of a speed check. A classic exam task is comparing the two values, here: average rate 4 on [1; 3], local rate f′(2) = 4 at the interval midpoint.

The derivative exists only if the limit of the difference quotient exists, from both sides with the same value. Three typical counterexamples: first, kinks like f(x) = |x| at x = 0, where the left-hand secant slope gives −1 and the right-hand one +1; no common limit, no tangent. Second, jump points: if f is discontinuous, there can be no derivative at all, differentiability requires continuity. Third, vertical tangents as for f(x) = ∛x at 0, where the secant slopes grow towards infinity. Conversely, continuity does not imply differentiability; |x| is the standard example of a continuous function that is not differentiable everywhere.