Short on time? This page is the recap. For the derivation, the graph and the worked examples, use the full study guide. BC-only material. You need Topic 9.1 first; the other prerequisites (quotient rule 2.9, chain rule 3.1, concavity 5.6) are on the Calculus BC roadmap.
Recap
- d²y/dx² is the derivative with respect to x of the slope dy/dx.
- In a parametric problem dy/dx is a function of t, so use the chain rule again: differentiate it with respect to t, then divide by dx/dt.
- Four steps: dx/dt and dy/dt → dy/dx (simplify!) → d/dt (dy/dx) → divide by dx/dt.
- The sign of d²y/dx² gives the concavity, exactly as for y = f(x).
Key relationships
| Quantity | Formula | Note |
|---|---|---|
| Slope | dy/dx = (dy/dt) ÷ (dx/dt) | dx/dt ≠ 0 |
| Second derivative | d²y/dx² = [d/dt (dy/dx)] ÷ (dx/dt) | Divide by dx/dt, not by d²x/dt² |
| Optional check | d²y/dx² = (x′y″ − y′x″) ÷ (x′)³ | Primes are t-derivatives |
| Concave up | d²y/dx² > 0 | Tangent line below the curve: underestimate |
| Concave down | d²y/dx² < 0 | Tangent line above the curve: overestimate |
The sign trap
If dx/dt < 0, the sign of d²y/dx² is the opposite of the sign of d/dt (dy/dx). Example: x = t² + 2t, y = t³ − 3t has d/dt (dy/dx) = 3/2 for all t, but d²y/dx² = 3/(4(t + 1)) is negative for t < −1.
Mistakes to avoid
- The shortcut (d²y/dt²) ÷ (d²x/dt²). For x = 2t, y = t² it gives 2 ÷ 0, but the true answer is ½.
- Forgetting the second division by dx/dt.
- Not simplifying dy/dx before differentiating it.
- Reading concavity from d/dt (dy/dx) without checking the sign of dx/dt.
- Chain-rule slips, for example d/dt [cos 2t] = −2 sin 2t.
- Over/under mixed up. Concave down means the estimate is too big.
Quick self-check
- For x = t², y = 4t³, find d²y/dx² at t = 1. (dy/dx = 6t; d/dt = 6; 6 ÷ 2t = 3/t, so 3)
- For x = 2t + 1, y = t² + t, find d²y/dx². (dy/dx = (2t + 1)/2; d/dt = 1; 1 ÷ 2 = ½)
- For x = cos t, y = sin t, is the curve concave up or down at t = π/2? (dy/dx = −cot t; d/dt = csc²t; ÷ (−sin t) gives −1/sin³t = −1, so concave down: the top of the circle)
Next: practice questions.