Unit 2: Differentiation — Definition & Fundamental Properties

The Difference Quotient

Defining the derivative as the limit of a difference quotient.

Advanced25 min lesson3 min readUpdated August 12, 2026Author not yet attributed

From a Secant Line to a Tangent Line

The slope of the line through two points on a curve is easy: rise over run. Before reading on: what happens to that secant line, and its slope, as the second point is dragged closer and closer to the first — is there a way to make that idea about the derivative precise?

Definition — The Difference Quotient and the Derivative

For two points on f's graph, (x, f(x)) and (x + h, f(x + h)), the difference quotient [f(x + h) − f(x)]/h is the slope of the secant line through them. The derivative f'(x) is defined as the limit of this difference quotient as h → 0 — the slope the secant line approaches as the second point slides into the first, which is exactly the slope of the tangent line at x.

f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

Worked Example — Finding a Derivative from the Definition

Find f'(x) for f(x) = x² using the limit definition. f(x + h) = (x + h)² = x² + 2xh + h², so f(x + h) − f(x) = 2xh + h². Divide by h: (2xh + h²)/h = 2x + h, valid for h ≠ 0 (exactly what a limit as h → 0 needs). Taking the limit: f'(x) = limₕ→0 (2x + h) = 2x. At x = 3, f'(3) = 6 — the instantaneous rate of change of f at x = 3 is 6.

Worked Example — A Derivative Requiring Fraction Algebra

Find f'(x) for f(x) = 1/x. f(x + h) − f(x) = 1/(x + h) − 1/x = [x − (x + h)] / [x(x + h)] = −h / [x(x + h)]. Divide by h: −1/[x(x + h)], valid for h ≠ 0. Taking the limit as h → 0 (so x + h → x): f'(x) = −1/x². At x = 2, f'(2) = −1/4 — the function 1/x is decreasing at x = 2, with an instantaneous rate of −0.25.

Derivative Explorer

f(x) =

x^3 - 3*x

Point of tangency

x = a1

Show tangent line

Trace mode

f(x), with tangent line at x = a

Show f''(x)Show f'''(x)

Derivative graph — linked to the graph above; panning or zooming either moves both

What's happening at x = a

At x = 1, f(x) ≈ -2 — the point (1, -2).
The derivative there is f'(1) ≈ 0: the slope of the tangent line, and the instantaneous rate of change of f at this exact point.
That slope is positive, so f is increasing at this point.
The second derivative is positive here, so f is concave up (curving upward) near this point.
Near x = 1, the tangent line y ≈ -2 + 0·(x − 1) is f's best straight-line approximation — the core idea a derivative captures.

Numerical derivative comparison

Forward, backward, and central difference estimates of f'(a) as the step size h shrinks — all three should converge toward the same value.
h	Forward	Backward	Central
1	4	-2	1
0.1	0.31	-0.29	0.01
0.01	0.0301	-0.0299	0.0001
0.001	0.003	-0.003	0

Analysis of f(x)

y-intercept: (0, 0)
x-intercepts: (-1.73, 0), (0, 0), (1.73, 0)
Extrema: local max at (-1, 2); local min at (1, -2)
Inflection points: (0, 0)
Vertical asymptotes: none found in view
Horizontal asymptotes: none found
Domain: all real numbers in view
Range (estimated): approximately [-970, 970]

Analysis of f'(x) — its roots are f's critical points

y-intercept: (0, -3)
x-intercepts: (-1, 0), (1, 0)
Extrema: local min at (0, -3)
Inflection points: none found in view
Vertical asymptotes: none found in view
Horizontal asymptotes: none found
Domain: all real numbers in view
Range (estimated): approximately [-3, 297]

Tip

When a difference quotient looks stuck at 0/0 as h → 0, that's expected, not a sign of a mistake — algebraically simplify first (expand, combine fractions, rationalize) until the troublesome h in the denominator cancels, the same 0/0-resolving instinct from Unit 1's limit laws.

Common Mistakes

Substituting h = 0 into the difference quotient before simplifying algebraically.
The unsimplified difference quotient is 0/0 at h = 0 — algebraic simplification has to happen first, so that the resulting expression is actually defined at h = 0 before the limit is evaluated.
Expanding (x + h)² as x² + h² instead of x² + 2xh + h².
(x + h)² requires the full binomial expansion, including the cross term 2xh — dropping it changes the entire computation and produces the wrong derivative.

Key Takeaways

The derivative f'(x) is the limit of the difference quotient [f(x+h) − f(x)]/h as h → 0 — the slope a secant line approaches as it becomes a tangent line.
Computing a derivative from the definition requires algebraically simplifying the difference quotient before taking the limit, since direct substitution of h = 0 gives 0/0.
f'(x) gives the instantaneous rate of change of f at x, generalizing the average rate of change (secant slope) to a single instant.

Summary

The limit definition is precise but slow. The next lesson asks a deeper question first — does every continuous function even have a derivative everywhere? — before Unit 2 develops faster rules for computing derivatives.

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