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Derivative rules from a table of values

Problem 2.1542 · easy

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

Quotient rule: h′ = (f′g − fg′)/g².
\[ 0 \]
Read the values from the table and substitute.✓ Proved

Answer \( h'(2) = 0 \)

Lines: 1 proved, 1 not checked. 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	Not checked	—	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: fail (error) — The solution fails to explicitly substitute the values from the table into the quotient rule formula, making the derivation opaque. It jumps directly to '0 = 0' without showing the intermediate calculation (-3*4 - (-4)*3) / 4^2, which is necessary to demonstrate how the numerator becomes zero.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to explicitly substitute the values from the table into the quotient rule formula, making the derivation opaque. It jumps directly to '0 = 0' without showing the intermediate calculation (-3*4 - (-4)*3) / 4^2, which is necessary to demonstrate how the numerator becomes zero.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to explicitly substitute the values from the table into the quotient rule formula. It jumps directly to the result '0 = 0' without showing the intermediate calculation (-3*4 - (-4)*3) / 4^2, making it impossible to verify the arithmetic or the application of the rule.
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.