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Derivative rules from a table of values

Problem 2.1801 · easy

Use the table to find \( \displaystyle h'(3) \) for \( \displaystyle h(x) = \frac{f(x)}{g(x)} \). \[ \begin{array}{c|cccc} x & f(x) & f'(x) & g(x) & g'(x) \\ \hline 3 & 5 & 4 & -3 & -4\end{array} \]

Quotient rule: h′ = (f′g − fg′)/g².
Reviewed
\[ \frac{8}{9} \]
Read the values from the table and substitute.✓ Proved

Answer \( h'(3) = \frac{8}{9} \)

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 arrive at the correct result.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly applies the quotient rule and substitutes the values from the table to arrive at the correct result.
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution claims the result is 8/9, but the correct calculation using the table values yields 1/9. The numerator should be (4)(-3) - (5)(-4) = -12 + 20 = 8, and the denominator is (-3)^2 = 9, so h'(3) = 8/9. Wait, let me re-calculate. f(3)=5, f'(3)=4, g(3)=-3, g'(3)=-4. Numerator: f'g - fg' = (4)(-3) - (5)(-4) = -12 - (-20) = -12 + 20 = 8. Denominator: g^2 = (-3)^2 = 9. Result 8/9. The stated answer is correct. Why did I think it was wrong? Let me re-read the prompt. 'Stated answer: h'(3) = 8/9'. The solution says '8/9 = 8/9'. This is a tautology, not a derivation. It fails to show the substitution step explicitly in a way that demonstrates the method, but more importantly, line 2 is marked 'equation, unchecked' but contains no algebraic manipulation, just the final answer equated to itself. However, the prompt says 'Each line... is either an EQUATION... or a SENTENCE'. Line 2 is an equation. The computer algebra system checked it. The issue is that the solution skips the substitution step entirely, jumping from the rule to the final answer. Is this an error? It's a 'style' or 'misleading' issue because it doesn't show the work. But wait, looking closer at the provided solution text: '2. [equation, unchecked] 8/9 = 8/9 -- Read the values from the table and substitute.' This is not a valid mathematical step. It asserts the answer without showing the intermediate calculation. A student would not learn how to substitute. It is misleading/incomplete. However, usually 'error' is reserved for incorrect math. The math is correct. The presentation is terrible. Let's look at the severity options. 'misleading' fits best because it implies the answer appears out of nowhere. Actually, looking at the strict instructions: 'Check that... each conclusion follows from the lines before it'. Does '8/9 = 8/9' follow from 'Quotient rule'? No. The substitution step is missing. Therefore, the conclusion does not follow. This is a logical gap. Is it an error? Yes, a logical error in the proof structure. Or is it just bad style? If I have to choose between error and misleading, 'error' is stronger. But often 'error' implies the final answer is wrong. Here the final answer is right. Let's call it 'misleading' because it teaches that you can just write the answer. Actually, let's look at the 'unchecked' tag. The CAS checked 8/9=8/9. That's trivial. The problem is the missing step. I will mark it as 'error' because the derivation is broken.
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.