Second derivatives
Problem 2.1155 · medium
Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = x^{3} - 4 \cos{\left(x \right)} \).
- \[ \frac{d}{d x} \left(x^{3} - 4 \cos{\left(x \right)}\right) = 3 x^{2} + 4 \sin{\left(x \right)} \]The first derivative.✓ Proved
- \[ \frac{d}{d x} \left(3 x^{2} + 4 \sin{\left(x \right)}\right) = 6 x + 4 \cos{\left(x \right)} \]Differentiate again.✓ Proved
Answer \( 6 x + 4 \cos{\left(x \right)} \)
✓ 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 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | matches a second difference quotient of f at x = 0.7 |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly computes the first and second derivatives. The steps are algebraically verified and the logic is sound.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly computes the first and second derivatives. The steps are algebraically verified and the logic is sound.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly computes the first and second derivatives. The steps are algebraically sound and follow standard differentiation rules.gpt-oss:20b: pass 2026-09-28
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/second_derivative, checked 2026-09-28 with SymPy 1.14.0.