Second derivatives of parametric curves
Problem 8.108 · 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 \( \displaystyle t = \frac{\pi}{4} \).
- \[ \frac{\frac{d}{d t} \tan{\left(t \right)}}{\frac{d}{d t} \sec{\left(t \right)}} = \frac{1}{\sin{\left(t \right)}} \]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
- \[ \left[\begin{matrix}\sqrt{2}\\-1\end{matrix}\right] \]At t = pi/4.✓ Proved
Answer \( \frac{dy}{dx} = \sqrt{2},\ \frac{d^2y}{dx^2} = -1 \)
✓ 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 and evaluates them at the specified point. The algebraic steps are verified and the final values match the stated answer.
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 steps are verified and the final values match the stated answer.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 and final values are correct.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.