Second derivatives of parametric curves
Problem 8.103 · medium
For \( \displaystyle x = - t^{2} + t \), \( \displaystyle y = t^{2} + 1 \), find \( \displaystyle \frac{dy}{dx} \) and \( \displaystyle \frac{d^2y}{dx^2} \) at \( \displaystyle t = 1 \).
- \[ \frac{\frac{d}{d t} \left(t^{2} + 1\right)}{\frac{d}{d t} \left(- t^{2} + t\right)} = - \frac{2 t}{2 t - 1} \]dy/dx = (dy/dt)/(dx/dt).✓ Proved
- \[ \frac{\frac{d}{d t} \left(- \frac{2 t}{2 t - 1}\right)}{\frac{d}{d t} \left(- t^{2} + t\right)} = \frac{2}{\left(1 - 2 t\right)^{3}} \]d²y/dx² = [d/dt (dy/dx)] / (dx/dt).✓ Proved
- \[ \left[\begin{matrix}-2\\-2\end{matrix}\right] \]At t = 1.✓ Proved
Answer \( \frac{dy}{dx} = -2,\ \frac{d^2y}{dx^2} = -2 \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.
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 parametric differentiation formulas for the first and second derivatives. The algebraic simplifications and final evaluation at t=1 are correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies the parametric differentiation formulas for the first and second derivatives. The algebraic simplifications and final evaluation at t=1 are correct.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies the formulas for first and second derivatives of parametric equations and evaluates them at t=1.gpt-oss:20b: pass 2026-10-05
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 2026-10-05 with SymPy 1.14.0.