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Second derivatives of parametric curves

Problem 8.139 · medium

For \( \displaystyle x = 2 \cos{\left(t \right)} \), \( \displaystyle y = 3 \sin{\left(t \right)} \), find \( \displaystyle \frac{dy}{dx} \) and \( \displaystyle \frac{d^2y}{dx^2} \) at \( \displaystyle t = \frac{\pi}{3} \).

\[ \frac{\frac{d}{d t} 3 \sin{\left(t \right)}}{\frac{d}{d t} 2 \cos{\left(t \right)}} = - \frac{3}{2 \tan{\left(t \right)}} \]
dy/dx = (dy/dt)/(dx/dt).✓ Proved
\[ \frac{\frac{d}{d t} \left(- \frac{3}{2 \tan{\left(t \right)}}\right)}{\frac{d}{d t} 2 \cos{\left(t \right)}} = - \frac{3}{4 \sin^{3}{\left(t \right)}} \]
d²y/dx² = [d/dt (dy/dx)] / (dx/dt).✓ Proved
\[ \left[\begin{matrix}\frac{\left(-1\right) \sqrt{3}}{2}\\- \frac{2 \sqrt{3}}{3}\end{matrix}\right] = \left[\begin{matrix}- \frac{\sqrt{3}}{2}\\- \frac{2 \sqrt{3}}{3}\end{matrix}\right] \]
At t = pi/3.✓ Proved

Answer \( \frac{dy}{dx} = - \frac{\sqrt{3}}{2},\ \frac{d^2y}{dx^2} = - \frac{2 \sqrt{3}}{3} \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the parametric differentiation formulas for the first and second derivatives. The algebraic simplifications and final evaluations at t = pi/3 are correct.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies the parametric differentiation formulas for the first and second derivatives. The algebraic simplifications and final evaluations at t = pi/3 are correct.
gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies the parametric differentiation formulas for the first and second derivatives. The algebraic simplifications and final evaluation at t = pi/3 are correct.

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.