How fast is it changing?

V1 — Average steepness on an interval.

Brand-new (steepness of change; optional prior: reading a function graph).

Tool gained: average rate of change = Δy/Δx = [f(b)−f(a)]/(b−a).

1. Principle. Over a stretch of input, rise divided by run is how steep the change was on average — same idea as a road grade.
2. Model. Average rate on [a, b] = [f(b) − f(a)] / (b − a).
3. Prediction / Answer. Temperature 12 °C at t = 1 h and 18 °C at t = 4 h → average +2 °C/h. Same tool for distance vs time (average speed).

V2 — Zoom the interval → slope at a point.

Builds on: average rate Δy/Δx (V1 tool).

Tool gained: derivative at a point f′(a) = lim_h→0 [f(a+h)−f(a)]/h (tangent slope).

1. Principle. Instantaneous steepness is what the average rates approach when the interval shrinks toward a single input — the secant becomes a tangent.
2. Model. f′(a) = lim_h→0 [f(a+h) − f(a)] / h, when the limit exists.
3. Prediction / Answer. For f(x) = x² at a = 3: [ (3+h)² − 9 ] / h = 6 + h → limit 6. So the tangent slope is 6 — transfer: speedometer reading at one clock instant.

V3 — Slope tool at every x + first shortcuts.

Builds on: f′(a) as limit of average rates (V2 tool).

Tool gained: f′ as a function; power/sum shortcuts: (xⁿ)′ = n xⁿ⁻¹, (cf)′ = c f′, (f+g)′ = f′+g′.

1. Principle. Once you can find the slope at one point, you can ask for the slope at every x — that answer is itself a function. Repeating the limit for powers earns shortcuts so you do not rebuild from scratch each time.
2. Model. If f(x) = xⁿ (n a positive integer), then f′(x) = n xⁿ⁻¹. Constants scale; sums differentiate term by term.
3. Prediction / Answer. f(x) = 4x³ − 5x + 2 → f′(x) = 12x² − 5. At x = 1 the slope is 7 — same limit idea, earned shortcut.