Skill goal: Discover the power, product, and quotient rules by checking them against the limit definition.
1 · Observe
Each tab shows a function whose derivative can be computed two ways: a secant slope from the limit definition, and the shortcut rule. Drag the point and watch them agree.
f(x) = x³ · candidate rule: f′(x) = 3x²
f(x) = (x² + 1)(x² − 1) = x⁴ − 1 · candidate rule: f′(x) = 4x³
f(x) = (x² + 1) / (x + 2), x ≠ −2 · candidate rule: f′(x) = (2x(x+2) − (x²+1)) / (x+2)²
Note: the draggable point is blocked within 0.05 of x = −2, where f is undefined.
2 · Manipulate
On each tab, drag the blue point left or right on the curve (range [−2, 2]). The limit-definition slope is computed numerically as the exact h → 0 limit (complex-step evaluation — no rounding error), and compared with the rule value. If the rule is right, the two numbers match to within 10⁻⁶ everywhere.
3 · Predict
Question: what is f′(x) for each function? Type your prediction, then reveal the rule steps.
4 · See
Live comparison — the definition and the rule agree everywhere you drag:
Each shows the largest |secant − rule| difference across 41 sampled points in [−2, 2].
5 · Explain
Power rule — binomial intuition: (a+h)³ = a³ + 3a²h + 3ah² + h³, so the difference quotient is 3a² + 3ah + h². As h → 0 the last two terms vanish, leaving 3a². The same binomial expansion gives (xⁿ)′ = nxⁿ⁻¹.
Product rule — “each factor takes a turn”: with u = x²+1 and v = x²−1,
f′ = u′v + uv′ = 2x(x²−1) + (x²+1)·2x = 4x³. One factor is differentiated while the other waits, then they swap.
Quotient rule — analogous, with a minus and the denominator squared: f′ = (u′v − uv′)/v² = (2x(x+2) − (x²+1))/(x+2)². The rule inherits the domain restriction x ≠ −2 from the function.
6 · Challenge
Compute each derivative at the given point. Answers are auto-checked (tolerance 0.01).
Score: 0 / 3