Second derivatives of parametric curves
Problem 8.135 · medium
For \( \displaystyle x = \frac{t^{2}}{2} \), \( \displaystyle y = \frac{t^{3}}{3} \), find \( \displaystyle \frac{dy}{dx} \) and \( \displaystyle \frac{d^2y}{dx^2} \) at \( \displaystyle t = 1 \).
- \[ \frac{\frac{d}{d t} \frac{t^{3}}{3}}{\frac{d}{d t} \frac{t^{2}}{2}} = t \]dy/dx = (dy/dt)/(dx/dt).✓ Proved
- \[ \frac{\frac{d}{d t} t}{\frac{d}{d t} \frac{t^{2}}{2}} = \frac{1}{t} \]d²y/dx² = [d/dt (dy/dx)] / (dx/dt).✓ Proved
- \[ \left[\begin{matrix}1\\1\end{matrix}\right] \]At t = 1.✓ Proved
Answer \( \frac{dy}{dx} = 1,\ \frac{d^2y}{dx^2} = 1 \)
Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | y found as a function of x by inverting x(t) numerically, then differenced |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies the formulas for first and second derivatives of parametric equations and evaluates them at t=1. The algebraic steps are verified and the final result matches the stated answer.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies the formulas for first and second derivatives of parametric equations and evaluates them at t=1. The algebraic steps are verified and the final result matches the stated answer.gpt-oss:20b: fail (error) 2026-10-06 — The solution miscomputes the derivatives: dy/dt of t^3/3 is t^2, not t, and consequently dy/dx is t^2, not t. The second derivative formula is applied to the wrong dy/dx, yielding 1/t instead of the correct value. The final numeric evaluation is therefore incorrect.qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies the parametric differentiation formulas for the first and second derivatives and evaluates them at t=1 to obtain the correct results.
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/parametric_second_derivative, checked 2026-10-06 with SymPy 1.14.0.