Unit 2: Differentiation — Definition & Fundamental Properties

Defining the Derivative

Defining the derivative as the limit of a difference quotient.

Advanced25 min lesson2 min readUpdated August 12, 2026Author not yet attributed

The Slope a Secant Line Approaches

Definition — The Derivative

f'(x) = limₕ→0 [f(x+h) − f(x)]/h — the slope the secant line through (x, f(x)) and (x+h, f(x+h)) approaches as h → 0, which is exactly the tangent line's slope. f is differentiable at x only where this limit exists as a finite number.

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 + 3x. f(x+h) − f(x) = [(x+h)² + 3(x+h)] − [x² + 3x] = 2xh + h² + 3h. Divide by h: 2x + h + 3. As h → 0: f'(x) = 2x + 3. At x = 2, f'(2) = 7 — the instantaneous rate of change there.

Worked Example — A Derivative Requiring Fraction Algebra

Find f'(x) for f(x) = 1/(x + 1). f(x+h) − f(x) = 1/(x+h+1) − 1/(x+1) = [(x+1) − (x+h+1)] / [(x+h+1)(x+1)] = −h/[(x+h+1)(x+1)]. Divide by h: −1/[(x+h+1)(x+1)]. As h → 0: f'(x) = −1/(x+1)².

This limit only exists where f is continuous — if limₕ→0 f(x+h) = f(x) failed, the difference quotient couldn't settle on a finite slope either. But continuity alone isn't enough: a corner, a cusp, or a vertical tangent can all leave a function continuous with no defined derivative at that exact point.

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

Simplify the difference quotient algebraically before taking the limit — direct substitution of h = 0 always gives 0/0 in the unsimplified form.

Common Mistakes

Substituting h = 0 before simplifying the difference quotient.
The unsimplified quotient is 0/0 at h = 0 — algebra has to remove that first.

Key Takeaways

f'(x) is the limit of the difference quotient as h → 0 — a secant slope becoming a tangent slope.
Computing a derivative from the definition requires simplifying algebraically before taking the limit.
Differentiability requires continuity, but continuity alone doesn't guarantee differentiability.

Summary

The limit definition is precise but slow. The next lesson develops the power, product, and quotient rules for computing derivatives quickly.

Track your progress

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