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Second derivatives

Problem 2.1271 · medium

Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \sqrt{x^{2} + 3 x + 3} \).

\[ \frac{d}{d x} \sqrt{x^{2} + 3 x + 3} = \frac{x + \frac{3}{2}}{\sqrt{x^{2} + 3 x + 3}} \]
The first derivative.✓ Proved
\[ \frac{d}{d x} \frac{x + \frac{3}{2}}{\sqrt{x^{2} + 3 x + 3}} = \frac{3}{4 \left(x^{2} + 3 x + 3\right)^{\frac{3}{2}}} \]
Differentiate again.✓ Proved

Answer \( \frac{3}{4 \left(x^{2} + 3 x + 3\right)^{\frac{3}{2}}} \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly computes the first and second derivatives. The algebraic steps are verified as correct, and the final result matches the stated answer.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly computes the first and second derivatives. The algebraic steps are verified as correct, and the final result matches the stated answer.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly computes the first and second derivatives. The algebraic simplification in the second step is correct, and the final result matches the stated answer.
gpt-oss:20b: pass 2026-09-29

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 2026-09-29 with SymPy 1.14.0.