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Second derivatives

Problem 2.743 · medium

Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \ln{\left(x^{2} + 4 \right)} \).

\[ \frac{d}{d x} \ln{\left(x^{2} + 4 \right)} = \frac{2 x}{x^{2} + 4} \]
The first derivative.✓ Proved
\[ \frac{d}{d x} \frac{2 x}{x^{2} + 4} = \frac{8 - 2 x^{2}}{x^{4} + 8 x^{2} + 16} \]
Differentiate again.✓ Proved

Answer \( \frac{2 \left(4 - x^{2}\right)}{x^{4} + 8 x^{2} + 16} \)

✓ 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. How we verify

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: pass
qwen3.6:27b-mlx: pass — The solution correctly computes the first and second derivatives. The algebraic simplification is verified by the CAS, and the steps are logically sound.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly computes the first and second derivatives. The algebraic simplification is verified by the CAS, and the steps are logically sound.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly computes the first and second derivatives. The algebraic simplification is verified as correct, and the steps logically follow.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly computes the first and second derivatives. The algebraic simplification in the second step is correct, as the denominator is the square of the term from the first derivative.
gpt-oss:20b: pass 2026-09-26

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.