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Second derivatives

Problem 2.799 · medium

Find \( \displaystyle \dfrac{d^2y}{dx^2} \) for \( \displaystyle y = \sqrt{x^{2} + 3 x + 4} \).

\[ \frac{d}{d x} \sqrt{x^{2} + 3 x + 4} = \frac{x + \frac{3}{2}}{\sqrt{x^{2} + 3 x + 4}} \]
The first derivative.✓ Proved
\[ \frac{d}{d x} \frac{x + \frac{3}{2}}{\sqrt{x^{2} + 3 x + 4}} = \frac{7}{4 \left(x^{2} + 3 x + 4\right)^{\frac{3}{2}}} \]
Differentiate again.✓ Proved

Answer \( \frac{7}{4 \left(x^{2} + 3 x + 4\right)^{\frac{3}{2}}} \)

✓ 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. 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 steps are verified, and the final result matches the stated answer.

Senior review claude-sonnet-5-5, 2026-10-13: pass — Both lines are correct and SymPy proved each. The first derivative is (2x+3)/(2√u). The quotient rule then gives (u − (x+3/2)²)/u^{3/2} = (7/4)/u^{3/2}, which matches the stated answer.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-13 — Both lines are correct and SymPy proved each. The first derivative is (2x+3)/(2√u). The quotient rule then gives (u − (x+3/2)²)/u^{3/2} = (7/4)/u^{3/2}, which matches the stated answer.
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly computes the first and second derivatives. The algebraic steps are verified, and the final result matches the stated answer.
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 steps are verified as correct, and the final result matches the stated answer.
gpt-oss:20b: fail (error) 2026-09-27 — The second derivative computed in the solution is incorrect; the correct result is (4x^2+12x+23)/(8(x^2+3x+4)^{3/2}), not 7/(4(x^2+3x+4)^{3/2}).
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly computes the first and second derivatives. The algebraic steps are verified, and the final result matches the stated answer.
gpt-oss:20b: fail (error) 2026-09-26 — The second derivative is not 7/(4*(x**2+3*x+4)**(3/2)); the correct result is -(2x^2+6x+1)/(2*(x**2+3*x+4)**(3/2)).

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.