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Daily Math Minute

Derivatives

The Difference Quotient

Defining the derivative as the limit of the difference quotient.

Advanced20 min lesson2 min readUpdated August 12, 2026Author not yet attributed

From a Secant Line to a Tangent Line

The slope between two points on a curve is easy: rise over run. Before reading on: what happens to that secant slope as the second point slides closer and closer to the first?

Definition — 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 secant line's slope. The derivative f'(x) is the limit of this quotient as h → 0 — the slope the secant approaches as it becomes a tangent line.
f′(x)=lim⁡h→0f(x+h)−f(x)hf'(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 − 4x. f(x+h) − f(x) = [(x+h)² − 4(x+h)] − [x² − 4x] = x² + 2xh + h² − 4x − 4h − x² + 4x = 2xh + h² − 4h. Divide by h: 2x + h − 4. As h → 0: f'(x) = 2x − 4. At x = 1, f'(1) = 2 − 4 = −2.

Derivative Explorer

Point of tangency
1

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

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

What's happening at x = a
  1. At x = 1, f(x) ≈ -2 — the point (1, -2).
  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.
  3. That slope is positive, so f is increasing at this point.
  4. The second derivative is positive here, so f is concave up (curving upward) near this point.
  5. 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.
hForwardBackwardCentral
14-21
0.10.31-0.290.01
0.010.0301-0.02990.0001
0.0010.003-0.0030
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

  • 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 produces the wrong derivative.

Key Takeaways

  • The derivative 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.

Summary

The next lesson interprets what this derivative actually means — both geometrically and in context.

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