The Limit Definition of the Derivative
Before the shortcut rules, power, product, chain, the derivative is a limit:
f′(x) = limh→0 [f(x + h) − f(x)] / h
The fraction inside, the difference quotient is the slope of a secant line through (x, f(x)) and a neighbor h away: an average rate over a shrinking interval. The limit watches those secant slopes as the neighbor slides in, and the number they settle on is the instantaneous rate: the slope of the tangent, the speedometer reading, the derivative. Every differentiation rule in the toolbox is a theorem about this limit; this page is where the rules come from, and where differentiability itself lives or dies.
How to Use This Calculator
Enter f and the point x. The calculator writes the difference quotient for your inputs, then tabulates it for h = ±0.1, ±0.01, down to ±0.00001, the limit process made visible as two columns of numbers marching toward each other. If both sides settle on a common value, it confirms the derivative (and cross-checks against the symbolic f′); if the sides disagree, it declares non-differentiability with the two one-sided slopes. The plot shows the geometry: a teal secant with h still finite, and the red tangent it's converging to.
Worked Example
f(x) = x² at x = 3, done symbolically the way every course requires once:
[f(3 + h) − f(3)]/h = [(9 + 6h + h²) − 9]/h = (6h + h²)/h = 6 + h → 6 as h → 0
The algebraic ritual, expand, cancel the 9's, factor out h, cancel it, then let h vanish, is the whole art: the cancellation must happen before the limit, because at h = 0 the original quotient is the meaningless 0/0. The calculator's table shows the same story numerically: 6.1, 6.01, 6.001 from the right; 5.9, 5.99, 5.999 from the left; both columns squeezing 6. The √x chip rehearses the other classic algebra (multiply by the conjugate), and 1/x the common-denominator route, the three standard hand techniques, all confirmable here.
When the Limit Fails: Non-Differentiability
The definition is also the diagnostic for where derivatives don't exist. The |x| chip runs it at 0: from the right, [|h| − 0]/h = h/h = +1 for every positive h; from the left, −1. Two one-sided limits, no agreement, no derivative, and the calculator says so explicitly. This is a corner: the graph is continuous but creased. Other failure modes the definition detects: cusps (x^(2/3) at 0, slopes blow up to ±∞ with a sign flip), vertical tangents (∛x at 0, both sides agree on ∞, a well-defined tangent line but no finite derivative), and outright discontinuities (differentiability requires continuity first, since the numerator must vanish with h). Continuous-but-not-differentiable is common; differentiable-but-not-continuous is impossible, an asymmetry worth internalizing early.
Common Mistakes to Avoid
- Substituting h = 0 immediately. That's 0/0, not an answer. The quotient must be simplified until the lone h downstairs cancels; only then does h → 0 make sense. This ordering is the technique.
- Expanding f(x + h) as f(x) + f(h). (x + h)² is x² + 2xh + h², not x² + h². Functions don't distribute over addition, the single most common algebra failure in these problems.
- Dropping the −f(x) term. The numerator is a difference; forgetting to subtract f(x) leaves a quotient that blows up instead of settling. If your table isn't converging, check the subtraction first.
- Checking only h > 0. The derivative needs both approaches to agree, one-sided tables miss corners entirely. (|x| at 0 looks perfectly differentiable if you only ever try positive h.)
Real-World Applications
The limit definition isn't just pedagogy, it's how derivatives are actually computed when no formula exists. Numerical differentiation in every simulation, spreadsheet, and sensor system is a finite-h difference quotient: velocity from GPS positions, acceleration from velocity logs, gradients in machine learning checked by "finite differences" against backpropagation (the standard debugging ritual of deep learning is literally this page's table). The h-tradeoff engineers manage, too large and the secant misrepresents the curve, too small and floating-point roundoff corrupts the subtraction, is visible in the calculator's own table if you imagine extending it far beyond 10⁻⁵.
Conceptually, the definition is load-bearing across science: marginal cost in economics is defined as this limit of cost differences; reaction rates, decay rates, and growth rates are all difference quotients of measurements pushed toward instantaneity; and physics's founding move, Newton and Leibniz's, in the 1660s–70s, was exactly the secant-to-tangent limit this tool animates. When later courses hand you f′ rules, they are compressing this limit; when research hands you data instead of formulas, you are back to computing it directly.
Frequently Asked Questions
Why does the definition use h instead of a second point b?
Pure convenience: b = x + h renames "second point approaches x" as "h approaches 0," making the algebra's target (cancel one factor of h) visible. The equivalent form limb→x [f(b) − f(x)]/(b − x) is identical and sometimes algebraically nicer (it factors differences like b³ − x³ beautifully).
What's the symmetric difference quotient, and why care?
[f(x + h) − f(x − h)]/(2h), slopes measured across the point. It converges faster (error ~h² vs ~h) and is what calculators use; but it can report a "derivative" at corners where none exists (it gives 0 for |x| at 0), which is why definitions use the one-sided-agreement version.
Why must the original quotient be 0/0 anyway?
Because f is continuous at x (or no derivative is possible), so f(x + h) − f(x) → 0 exactly as h → 0. The derivative is the resolution of that particular 0/0: how fast the top vanishes relative to the bottom. Every derivative is an indeterminate form tamed.
How do the derivative rules follow from this definition?
Each is the limit computed once in general: the power rule from expanding (x + h)ⁿ, the product rule from an add-and-subtract trick in the numerator, the chain rule from composing quotients. A course proves them here, then never uses the definition again, this page is the museum of that founding computation.
Can a function be differentiable exactly once, not twice?
Yes: x·|x| has derivative 2|x| everywhere (check both sides at 0!), but 2|x| itself has a corner at 0, so f″(0) fails. Differentiability is a ladder climbed one limit at a time, and each rung can be the last.
What does the table do when the derivative is huge?
For steep functions the quotients settle slowly and roundoff bites earlier, try 1/x at 0.01. Numerically, extreme slopes compress the reliable h-window from both ends, a genuine limitation the symbolic cross-check (computed by the engine's rules) is immune to.