Derivative rules from a table of values
Problem 2.2503 · easy
Use the table to find \( \displaystyle h'(3) \) for \( \displaystyle h(x) = \frac{f(x)}{g(x)} \). \[ \begin{array}{c|cccc} x & f(x) & f'(x) & g(x) & g'(x) \\ \hline 3 & -2 & 3 & 3 & 3\end{array} \]
- Quotient rule: h′ = (f′g − fg′)/g².
- \[ \frac{2 \cdot 3 + 3 \cdot 3}{3^{2}} = \frac{5}{3} \]Read the values from the table and substitute.✓ Proved
Answer \( h'(3) = \frac{5}{3} \)
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. How we verify
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 substitution in step 2 is incorrect. The table gives f(3) = -2, but the calculation uses 2 (implied by 2*3). The correct numerator should be (3*3 - (-2)*3) = 15, leading to h'(3) = 15/9 = 5/3. While the final answer is coincidentally correct due to the specific numbers, the intermediate step shown is mathematically false and demonstrates a sign error in applying the quotient rule.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-26 — The substitution in step 2 is incorrect. The table gives f(3) = -2, but the calculation uses 2 (implied by 2*3). The correct numerator should be (3*3 - (-2)*3) = 15, leading to h'(3) = 15/9 = 5/3. While the final answer is coincidentally correct due to the specific numbers, the intermediate step shown is mathematically false and demonstrates a sign error in applying the quotient rule.gpt-oss:20b: pass 2026-10-26qwen3.6:27b-mlx: inconclusive 2026-10-25 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The substitution in step 2 is incorrect. The table gives f(3) = -2, but the calculation uses 2 (positive). The correct numerator should be (3*3 - (-gpt-oss:20b: pass 2026-10-25qwen3.6:27b-mlx: fail (error) 2026-10-25 — The substitution in step 2 uses f(3)=2 instead of the table value f(3)=-2. The correct numerator should be (3)(3) - (-2)(3) = 15, yielding h'(3) = 15/9 = 5/3. While the final answer is coincidentally correct due to a sign error cancellation, the intermediate calculation shown is factually wrong.gpt-oss:20b: fail (misleading) 2026-10-25 — The solution substitutes 2 for f(3) instead of –2 and adds 3·3 rather than subtracting f(3)·g′(3). Although the arithmetic happens to give the correct result, the reasoning is incorrect and would mislead a student about how to apply the quotient rule.
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 2026-10-26 with SymPy 1.14.0.