Second derivatives of parametric curves
Problem 8.104 · medium
For \( \displaystyle x = \sec{\left(t \right)} \), \( \displaystyle y = \tan{\left(t \right)} \), find \( \displaystyle \frac{dy}{dx} \) and \( \displaystyle \frac{d^2y}{dx^2} \) at \( \displaystyle t = \frac{\pi}{3} \).
- \[ \frac{\frac{d}{d t} \tan{\left(t \right)}}{\frac{d}{d t} \sec{\left(t \right)}} = \frac{1}{\sin{\left(t \right)}} \]dy/dx = (dy/dt)/(dx/dt).✓ Proved
- \[ \frac{\frac{d}{d t} \frac{1}{\sin{\left(t \right)}}}{\frac{d}{d t} \sec{\left(t \right)}} = - \frac{1}{\tan^{3}{\left(t \right)}} \]d²y/dx² = [d/dt (dy/dx)] / (dx/dt).✓ Proved
- \[ \left[\begin{matrix}\frac{2 \sqrt{3}}{3}\\\frac{\left(-1\right) \sqrt{3}}{9}\end{matrix}\right] = \left[\begin{matrix}\frac{2 \sqrt{3}}{3}\\- \frac{\sqrt{3}}{9}\end{matrix}\right] \]At t = pi/3.✓ Proved
Answer \( \frac{dy}{dx} = \frac{2 \sqrt{3}}{3},\ \frac{d^2y}{dx^2} = - \frac{\sqrt{3}}{9} \)
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 parametric differentiation formulas and evaluates them at the specified point. The algebraic simplifications are correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies the parametric differentiation formulas and evaluates them at the specified point. The algebraic simplifications are correct.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies the parametric differentiation formulas and evaluates them at the specified point. The algebraic simplifications are correct.gpt-oss:20b: fail (error) 2026-10-05 — The second sentence incorrectly computes the derivative of 1/sin(t); it should be -cos(t)/sin^2(t), not -1/tan^3(t).
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.