Derivative rules from a table of values
Problem 2.1800 · easy
Use the table to find \( \displaystyle h'(0) \) for \( \displaystyle h(x) = \frac{f(x)}{g(x)} \). \[ \begin{array}{c|cccc} x & f(x) & f'(x) & g(x) & g'(x) \\ \hline 0 & 4 & -4 & 2 & 2\end{array} \]
- Quotient rule: h′ = (f′g − fg′)/g².
- \[ -4 \]Read the values from the table and substitute.✓ Proved
Answer \( h'(0) = -4 \)
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: passqwen3.6:27b-mlx: fail (error) — The solution fails to substitute the specific values from the table into the quotient rule formula. It merely states the rule and then asserts an incorrect equality (-4 = -4) without showing the calculation (e.g., (-4*2 - 4*2)/2^2 = -16/4 = -4). The final answer is also incorrect; the correct value is -4, but the derivation shown is nonsensical and does not lead to the answer.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to substitute the specific values from the table into the quotient rule formula. It merely states the rule and then asserts an incorrect equality (-4 = -4) without showing the calculation (e.g., (-4*2 - 4*2)/2^2 = -16/4 = -4). The final answer is also incorrect; the correct value is -4, but the derivation shown is nonsensical and does not lead to the answer.qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to substitute the values into the quotient rule formula. It merely states the result equals itself (-4 = -4) without showing the calculation (f'(0)g(0) - f(0)g'(0)) / g(0)^2, which is (-4*2 - 4*2) / 2^2 = -16/4 = -4.gpt-oss:20b: pass 2026-10-07
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 2026-10-07 with SymPy 1.14.0.