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Lesson 05 · Core idea one · 9 min

Derivatives

The derivative is the slope of a curve at a single point — the instantaneous rate of change. It answers “how fast, right now?” for anything a function can describe.

Build the intuition

From average to instant

Pick a point on a curve and a second point nearby. The line through both has a measurable slope — an average rate. Now slide the second point closer. The slopes settle toward one number: the slope of the tangent line. That number is the derivative.

f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

The tangent line

The tangent just kisses the curve, matching its direction at exactly one point. Zoom in far enough and the curve becomes indistinguishable from its tangent — curves are secretly straight up close, and the derivative is the slope of that hidden straightness.

The derivative is itself a function

Every point has its own slope, so the slopes form a new function: f′ (read “f prime”). For f(x) = x², the slope at any x turns out to be exactly 2x. One formula, every tangent.

f(x)=x2  ⇒  f′(x)=2xf(x) = x^2 \;\Rightarrow\; f'(x) = 2x

See it move

InteractiveFrom average to instant
1.6
Average rate between the two points: 3.6. As the gap shrinks toward 0, it closes in on 2 — the instant rate at x = 1.

Shrink the gap and watch the orange average-rate line settle onto the dashed tangent. The derivative is the destination of that journey.

A worked example

The slope of x² at x = 3

  1. Average slope between 3 and 3 + h:

    (3+h)2−9h=6h+h2h=6+h\frac{(3+h)^2 - 9}{h} = \frac{6h + h^2}{h} = 6 + h

  2. Shrink the gap:

    lim⁡h→0 (6+h)=6\lim_{h \to 0}\,(6 + h) = 6

  3. So f′(3) = 6 — and the h cancelled before we ever divided by zero. That cancel-then-shrink move is the heart of every derivative.

The formula, earned

f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}

The definition: average slope over a shrinking gap.

ddx x2=2x\frac{d}{dx}\,x^2 = 2x

Your first derivative — the parabola's slope formula.

Out in the world

Your speedometer

A speedometer reports the derivative of your position — not your trip average. “80 km/h right now” is a limit your car computes continuously. You've trusted derivatives every ride of your life.

Common confusion, cleared

“The derivative is a single number for a function.”

It's a whole new function — a slope for every point. f′(3) = 6 is one reading; f′(x) = 2x is the full instrument.

“dy/dx is a fraction.”

It's notation for a limit of fractions. It often behaves like a fraction (which is why the notation is beloved), but treat that as a happy pattern, not a literal division.

Check yourself

PracticeQuick check

  1. A curve is momentarily flat at a point. Its derivative there is…

  2. f′(2) = −3 means that near x = 2, f is…

Recap

  • Derivative = instantaneous rate = tangent slope.
  • Built as a limit of average slopes over shrinking gaps.
  • The derivative of a function is itself a function.