Derivative rules from a table of values
Problem 2.1647 · easy
Use the table to find \( \displaystyle h'(0) \) for \( \displaystyle h(x) = \frac{f(x)}{g(x)} \). \[ \begin{array}{c|cccc} x & f(x) & f'(x) & g(x) & g'(x) \\ \hline 0 & 4 & 1 & -1 & 3\end{array} \]
- Quotient rule: h′ = (f′g − fg′)/g².
- \[ -13 \]Read the values from the table and substitute.✓ Proved
Answer \( h'(0) = -13 \)
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 claims to substitute values from the table but fails to show the substitution step, making it impossible to verify the arithmetic. Furthermore, the calculation is incorrect: h'(0) = (1*(-1) - 4*3) / (-1)^2 = -13 / 1 = -13. Wait, the calculation is actually correct. Let me re-read the prompt constraints. 'Each line... is either an EQUATION... or a SENTENCE'. Line 2 is marked [equation, proved]. The prompt says 'do not re-check the algebra of a line marked proved'. However, the sentence in line 1 is just stating the rule. The 'substitution' is described in the text of line 2 but not shown as an equation. The verdict should be based on whether the *sentences* are correct. The sentence in line 2 says 'Read the values... and substitute' but presents '-13 = -13'. This is not a valid mathematical derivation step. It skips the crucial step of showing the substitution: (1*(-1) - 4*3)/(-1)^2 = -13. A student would not learn how to apply the rule. It is misleading/error because it hides the work. Actually, looking closely, the result -13 is correct. The issue is the presentation. Is it an error? The prompt asks to find what is wrong. The 'equation' line is technically 'proved' by CAS, so I shouldn't check the algebra. But the *sentence* part of line 2 is 'Read the values from the table and substitute.' This is a valid instruction. The equation '-13 = -13' is the result. The problem is that the intermediate step is missing. Is this 'style' or 'error'? If I strictly follow 'do not re-check algebra', and the final answer is correct, is it pass? No, because the solution is incomplete. It doesn't show *how* the values were used. A student needs to see (f'(0)g(0) - f(0)g'(0)) / g(0)^2. Skipping this makes it misleading about the process. I will mark it as misleading because it obscures the application of the quotient rule.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution claims to substitute values from the table but fails to show the substitution step, making it impossible to verify the arithmetic. Furthermore, the calculation is incorrect: h'(0) = (1*(-1) - 4*3) / (-1)^2 = -13 / 1 = -13. Wait, the calculation is actually correct. Let me re-read the prompt constraints. 'Each line... is either an EQUATION... or a SENTENCE'. Line 2 is marked [equation, proved]. The prompt says 'do not re-check the algebra of a line marked proved'. However, the sentence in line 1 is just stating the rule. The 'substitution' is described in the text of line 2 but not shown as an equation. The verdict should be based on whether the *sentences* are correct. The sentence in line 2 says 'Read the values... and substitute' but presents '-13 = -13'. This is not a valid mathematical derivation step. It skips the crucial step of showing the substitution: (1*(-1) - 4*3)/(-1)^2 = -13. A student would not learn how to apply the rule. It is misleading/error because it hides the work. Actually, looking closely, the result -13 is correct. The issue is the presentation. Is it an error? The prompt asks to find what is wrong. The 'equation' line is technically 'proved' by CAS, so I shouldn't check the algebra. But the *sentence* part of line 2 is 'Read the values from the table and substitute.' This is a valid instruction. The equation '-13 = -13' is the result. The problem is that the intermediate step is missing. Is this 'style' or 'error'? If I strictly follow 'do not re-check algebra', and the final answer is correct, is it pass? No, because the solution is incomplete. It doesn't show *how* the values were used. A student needs to see (f'(0)g(0) - f(0)g'(0)) / g(0)^2. Skipping this makes it misleading about the process. I will mark it as misleading because it obscures the application of the quotient rule.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to show the substitution of the specific values from the table (f(0)=4, f'(0)=1, g(0)=-1, g'(0)=3) into the quotient rule formula. It merely states the final result without demonstrating the calculation, making it impossible to verify if the arithmetic was performed correctly or if the correct values were used.gpt-oss:20b: pass 2026-10-05
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 2026-10-05 with SymPy 1.14.0.