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Derivative rules from a table of values

Problem 2.1539 · easy

Use the table to find \( \displaystyle h'(2) \) for \( \displaystyle h(x) = \frac{f(x)}{g(x)} \). \[ \begin{array}{c|cccc} x & f(x) & f'(x) & g(x) & g'(x) \\ \hline 2 & -4 & 1 & -3 & -2\end{array} \]

Quotient rule: h′ = (f′g − fg′)/g².
Reviewed
\[ - \frac{11}{9} \]
Read the values from the table and substitute.✓ Proved

Answer \( h'(2) = - \frac{11}{9} \)

Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
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	differentiated concrete polynomials built to match the table

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the quotient rule and substitutes the values from the table to obtain the correct result.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the quotient rule and substitutes the values from the table to obtain the correct result.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to show the substitution of values from the table into the quotient rule formula. It jumps directly to the final answer without demonstrating the calculation steps (e.g., substituting f(2)=-4, f'(2)=1, g(2)=-3, g'(2)=-2), making it impossible to verify the arithmetic or logic.
gpt-oss:20b: pass 2026-10-04

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/derivative_from_table, checked 2026-10-04 with SymPy 1.14.0.