Teaching objectives
The derivative is the most important idea in calculus, and it is almost always introduced all at once as a hard-to-digest limit. This lab builds it the other way round: first the intuition —"how fast does this function rise?"— and only then the formula. Everything happens on a 1:1 grid, so the student can estimate slopes by eye, counting squares, before computing anything. Across 32 rooms it walks the full path, from tapping where a curve rises fastest to differentiating with the chain rule, understanding how a computer approximates a derivative, and solving optimisation problems. It is aimed at High school and University.
What you learn
The journey goes from the visual to the symbolic and back to the applied. First the derivative as "rate of rise", then as the slope of the tangent, then as a limit, then as something a computer calculates, and finally as a tool for calculation and optimisation.
- See the derivative as instantaneous speed (a car's speedometer) and recognise where a function rises, falls or flattens, reading the sign of the derivative off its graph (rooms 1 to 4).
- Build the derivative by dragging the point of tangency, and match f with its derivative in both directions (rooms 5 and 6).
- Review the slope of a line and understand it as tan(angle) on the grid (room 7).
- Move from the average rate of change (secant) to the instantaneous one (tangent) by taking h→0, and see that every smooth curve is almost a line if you zoom in enough (rooms 8 to 11).
- Compute derivatives from the definition as a limit for the line, x², x³, 1/x and √x, reaching the power rule; recognise points where there is no derivative (rooms 12 to 16).
- Understand how a computer approximates a derivative with a fixed step h, and why the symmetric derivative is much more accurate than the forward one (rooms 17 and 18).
- Differentiate polynomials and elementary functions and apply the product, quotient and chain rules with visual justifications, testing memory and recognition speed in two against-the-clock tests (rooms 19 to 26).
- Write the tangent line, recognise concave and convex curves and the inflection point, study the shape of the curve with f′ and f″, and solve two geometric optimisation problems (rooms 27 to 31).
- Close with a no-challenge bonus room: a curl noise field generated from partial derivatives, showing that the derivative also creates generative art (room 32).
Key mathematical ideas
- The derivative measures how fast a function changes: where it is positive the function rises, where negative it falls, and where it vanishes there is a possible maximum or minimum.
- The derivative at a point is the slope of the tangent line, the limit of secant slopes as the second point approaches the first: f′(x₀) = lim_{h→0} (f(x₀+h)−f(x₀))/h.
- Local linearity: a differentiable function, seen up close, is indistinguishable from a line —and that line is its tangent—, the idea that explains why the derivative is useful for approximation.
- From the definition we obtain the elementary derivatives; the power rule (xⁿ)′ = n·xⁿ⁻¹ summarises them and extends to fractional and negative exponents.
- Not every continuous function is differentiable: at a peak (like |x|) the two "half-tangents" have different slopes and there is no single tangent.
- A computer doesn't compute limits: it approximates. The forward difference has an error proportional to h; the symmetric (centered) difference has an error proportional to h², so it is much more accurate for the same step.
- The rules of differentiation —product, quotient and chain— let us differentiate any combination of known functions.
- The sign of f′ describes increase/decrease and that of f″ describes concavity; together they give the shape of the graph, with extrema where f′=0 and inflection points where f″ changes sign.
- In optimisation, maxima and minima are found by solving f′=0; the same minimum-time principle explains Snell's law of refraction.
Room-by-room contents
Room 1 · The car and the speedometer
A car drives down a road while the position-time graph is drawn live; below, a speedometer needle shows the instantaneous speed. The student plays/pauses the animation and spots the fastest stretch and the stopped stretch.
Student tasks
- Watch the run and relate the curve’s steepness to the needle.
- Pause when the car is fastest and when it is stopped.
Room 2 · How fast does it rise?
The student first taps where the curve rises fastest and the two points where it neither rises nor falls. Then, with a play/pause button, an animation runs a point along f and, synchronised below, its derivative f′, with live speed captions.
Student tasks
- Tap the steepest point and the two flat points.
- Play the animation of f and f′ and read the speed captions.
Room 3 · The sign of the derivative
On a cubic with one minimum and one maximum, the student paints each stretch green (rising, f′>0) or yellow (falling, f′<0) and checks. It cements the link between monotonicity and the sign of the derivative.
Student tasks
- Mark the three stretches as rising or falling.
- Relate rising ↔ f′>0 and falling ↔ f′<0.
Room 4 · Peaks and valleys
On a function with two minima and one maximum, the student taps the three points where the tangent is horizontal (f′=0). A horizontal tangent appears when each is found.
Student tasks
- Locate the peaks and troughs of the curve.
- Tap the three points where f′=0.
Room 5 · Trace the derivative yourself
The student drags the point of tangency along f; the panel below leaves a trail with the slope value, drawing f′ with their own finger. Once the curve is covered, the real f′ is overlaid to compare.
Student tasks
- Drag along the whole curve tracing the tangent point.
- Compare the traced trail with the real graph of f′.
Room 6 · Match f and f′
Four rounds alternating direction: twice "given f, choose f′" (parabola, increasing cubic) and twice "given f′, choose f" (constant, the line −x).
Student tasks
- Analyse where f rises and falls (or the sign of f′).
- Complete the four rounds.
Room 7 · Slope: step and angle
First a slope slider shows the step "1 to the right, m up", with two identification questions. Then an angle slider θ rotates the line and shows m = tan θ, with two more questions.
Student tasks
- Read the slope as the step height.
- Vary the angle and check that m = tan θ.
Room 8 · Average rate of change
On y = x²/2 the secant (dashed gray) between two points is drawn with the Δy/Δx triangle. The student computes the average rate of change over two intervals.
Student tasks
- Compute Δy and Δx between the given points.
- Obtain the average rate of change (slope of the secant).
Room 9 · From secant to tangent
An h slider brings the second point toward the first. Past secants fade into a trail as they rotate into the red tangent; a large number shows the slope settling at 2, with a success animation.
Student tasks
- Reduce h and watch the trail of secants rotate.
- Check that the slope tends to 2 as h→0.
Room 10 · Zoom in: the curve is almost a line
A zoom slider (1× to 200×) zooms in on a point of a smooth curve until it looks like a line; checking it overlays the tangent. It repeats at a second point with more curvature.
Student tasks
- Zoom in to at least 40× at each point.
- Check that the line matches the tangent.
Room 11 · Estimate the slope
On a smooth cubic, with the tangent drawn as a hint, the student estimates the slope by eye at three points, counting grid squares and choosing the approximate value.
Student tasks
- Count how much the tangent rises per step to the right.
- Pick the correct estimate at the three points.
Room 12 · The definition
We present f′(x₀) = lim (f(x₀+h)−f(x₀))/h. A table with ever smaller h shows the quotient approaching 6 for f=x² and x₀=3; hovering over a row highlights its matching secant on the graph.
Student tasks
- Follow the table and see which number the quotient tends to.
- Write the value of the limit (f′(3)=6).
Room 13 · Derivative of a line
Limit proof that the derivative of a·x+b is a, at every point. Three exercises: 3x−5, −2x+7 and a constant (derivative 0).
Student tasks
- Follow the limit proof.
- Compute f′ for the three lines.
Room 14 · Derivative of x²
Expansion (x+h)²−x² = 2xh+h² ⇒ f′(x)=2x. The student evaluates f′(3), f′(−5) and finds where f′=0 (the vertex).
Student tasks
- Follow the limit expansion to 2x.
- Evaluate f′ at the requested points.
Room 15 · Three gates: x³, 1/x, √x
The student chooses the order to open three gates, each with its own limit and algebraic trick (binomial, common denominator, conjugate). Finishing all three unlocks the power rule xⁿ → n·xⁿ⁻¹.
Student tasks
- Complete the three limits in any order.
- Apply the power rule, including fractional and negative exponents.
Room 16 · Where there is no derivative
Three curves (|x|, ∛x and x²) with sliders bringing a point closer from the left and the right (the left one in magnitude, matching the right one); two large numbers show both slopes. It first explains that the derivative exists when the one-sided limits coincide and are finite, and one question per curve decides whether a single tangent exists, with the conclusion in large text and an "Understood" button.
Student tasks
- Compare the slope from the left and from the right on each curve.
- Decide whether a single tangent exists.
Room 17 · The derivative a computer calculates
With a fixed-h slider, the student sees the forward secant next to the real tangent and an error number that shrinks as h drops; a mini chart shows that too small an h makes the error grow again.
Student tasks
- Move h and watch the secant approach the tangent.
- Find the h with the least total error.
Room 18 · Forward vs. symmetric
On the same curve, the real tangent, the forward secant and the symmetric (centered) secant are compared, with a live numeric table. Final challenge: match with the forward method the precision the symmetric one gets at h=0.5.
Student tasks
- Compare which secant tracks the tangent better as h changes.
- Find an h for the forward method matching the symmetric one’s precision.
Room 19 · Power, constant and sum
Linearity of the derivative: (c·u)′=c·u′, (u+v)′=u′+v′, (c)′=0. Several derivatives of polynomials and one equation f′=0.
Student tasks
- Differentiate polynomials term by term.
- Evaluate f′ and solve f′=0.
Room 20 · Sine and cosine
The student traces the derivative of sin x by dragging; completing the trace reveals it is cos x. Then they mark on sine itself the points where it decreases fastest and the points where it neither rises nor falls. Everything is hidden and the whole process repeats with cos x: trace its derivative (−sin x), decide whether it is positive or negative on (0,π), and mark where cosine rises fastest.
Student tasks
- Trace the derivative of sin x by dragging and mark its key points.
- Repeat the whole process with cos x.
Room 21 · Exponential and logarithm
The student traces the derivative of eˣ by dragging and discovers it lands on the curve itself. Then they trace the derivative of ln x and identify which known function it resembles (multiple choice: x², √x, 1/x, 1/x²). It closes with two evaluation exercises.
Student tasks
- Trace the derivative of eˣ and check it is itself.
- Trace the derivative of ln x, identify it as 1/x and compute at a point.
Room 22 · Speed test: basic derivatives
The table of the 10 basic derivatives is shown (x², x³, x⁵, 1/x, √x, ln x, eˣ, sin x, cos x, tan x) and, once the student is ready, a countdown starts. Each round asks to pick the correct derivative among 5 options before time runs out: a correct answer cuts the time limit by 300 ms and adds 1 point, a mistake adds 400 ms and subtracts 1 point. All 10 rounds must be finished with a positive score.
Student tasks
- Memorize the table of basic derivatives.
- Clear the 10 against-the-clock rounds with a positive score.
Room 23 · The product rule
A rectangle with sides u and v grows via a Δ slider, showing the two strips u·Δv and v·Δu and the negligible corner Δu·Δv from which the area increment and (f·g)′ = f′·g + f·g′ come. Exercises use exponential or logarithm times a polynomial, so they can't be solved by expanding parentheses.
Student tasks
- Increase the rectangle’s sides and identify the strips and the corner.
- Differentiate each factor and combine with the rule.
Room 24 · The quotient rule
We state (f/g)′ = (f′·g − f·g′)/g² and apply it in three exercises with algebraic fractions, with a warning about the order of the subtraction.
Student tasks
- Apply the quotient rule step by step.
- Evaluate h′ at the requested point.
Room 25 · The chain rule
A gear animation (x drives g, g drives f) shows that speeds multiply: (f∘g)′ = f′(g(x))·g′(x). Three application exercises.
Student tasks
- Watch the gear animation and deduce that speeds multiply.
- Differentiate the three proposed cases.
Room 26 · Speed test: chain rule
A countdown and 10 against-the-clock rounds: for each function (cos(3x), ln(5x+1), √(2−x), e^(x²), (3−x²)¹⁰ and five more) the student picks its derivative among 5 very similar options before time runs out. Same scoring and timing system as room 22: the round must end with a positive score.
Student tasks
- Recognize at a glance the derivative of functions needing the chain rule.
- Clear the 10 against-the-clock rounds with a positive score.
Room 27 · Tutorial: the tangent line
The full worked example (y = x²/2, x₀=2) is modelled first, then the student repeats alone with a negative-slope function: computes f(x₀), f′(x₀) and places the tangency point on the graph before seeing the line.
Student tasks
- Follow the modelled exercise.
- Compute f(x₀) and f′(x₀) for the second one, place the point and see the tangent.
Room 28 · Concave and convex
Across two stretches of the same curve, the student compares the curve with the segment (secant) joining its endpoints: concave if the curve stays above, convex if it stays below. This is linked to the second derivative, and the student is asked to locate the inflection point by eye, where concavity changes.
Student tasks
- Classify the two stretches as concave or convex by comparing with the segment.
- Locate the inflection point by eye.
Room 29 · The shape of the curve
Three stacked, synced panels —f, f′ and f″— with a vertical line running through them. Challenge 1: tap on f′ where it crosses 0 (extrema). Challenge 2, unlocked next: tap on f″ where it crosses 0 (inflection). The sign change of f′ marking extrema and of f″ marking inflection is explained, then three sign questions close the room.
Student tasks
- Challenge 1: mark the two extrema on the f′ panel.
- Challenge 2: mark the inflection point on the f″ panel and answer the sign questions.
Room 30 · The box of maximum volume
From a 12×12 sheet the student cuts squares of side x at the corners (slider, no visible numbers) and folds; the box and the curve V(x) are shown. They first pick the correct formula for V′(x) among four options, and only then set it to 0 and solve, finding the optimal cut x=2, with V=128 cm³.
Student tasks
- Pick the correct formula for V′(x).
- Set it to 0, solve and check the maximum V(2)=128.
Room 31 · The lifeguard
A lifeguard runs across the sand at 6 m/s and swims at 2 m/s: time is not minimised in a straight line. Calling x the horizontal distance to the entry point, the student picks among 4 options the distance covered on the sand, then in the water. The formula T(x) and T′(x) are revealed, and using the slider the student must precisely find, to the decimal, the x that minimises the time. On success, that condition is revealed to be Snell’s law of refraction (Fermat’s principle), with a schematic of a light ray.
Student tasks
- Identify the expressions for the two distances in terms of x.
- Precisely find the x that minimises T(x) and relate it to Snell’s law.
Room 32 · Generative art with derivatives
A no-challenge bonus room: a curl noise field, generated from the partial derivatives of a potential function ψ(x,y), draws animated swirls that never have sources or sinks, with a button to regenerate the pattern.
Student tasks
- Observe how partial derivatives generate a flow with no sources or sinks.
- Regenerate the pattern and explore the result.
Rooms to project
The most striking ones to show and discuss in class.