Skill: derivative-rules

Derivative Rules Lab

The three rules that differentiate almost everything: the product rule for multiplied chunks, the quotient rule for fractions, and the chain rule for functions inside functions. Master these and every derivative becomes bookkeeping.

1 Chain-rule machine

Explore f(x) = (ax + b)ⁿ. Drag the sliders and watch the function (blue), its derivative (teal), and the tangent line (gold) move together. The readout breaks f′(x₀) into outer′ × inner′.

a (inner slope): 2

b (inner shift): 1

n (outer power): 3

x₀ (tangent point): 1.0

f(x₀) = –

f′(x₀) = –

outer′ × inner′ = –

2 What to notice

The teal curve crosses zero exactly where the blue curve flattens — the derivative is zero at horizontal tangents.
Raise n: the outer derivative grows a power, and the teal curve steepens faster than the blue one.
Change a: the inner derivative scales everything — the readout’s second factor changes, and the whole picture stretches.
The gold tangent always has slope f′(x₀), matching the teal curve’s height at x₀.

Use the stepper Practice