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Derivative rules from a table of values

Problem 2.2502 · easy

Use the table to find \( \displaystyle h'(2) \) for \( \displaystyle h(x) = \frac{f(x)}{g(x)} \). \[ \begin{array}{c|cccc} x & f(x) & f'(x) & g(x) & g'(x) \\ \hline 2 & 4 & 4 & 5 & 1\end{array} \]

Quotient rule: h′ = (f′g − fg′)/g².
\[ \frac{\left(-4\right) 1 + 4 \cdot 5}{5^{2}} = \frac{16}{25} \]
Read the values from the table and substitute.✓ Proved

Answer \( h'(2) = \frac{16}{25} \)

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: pass
qwen3.6:27b-mlx: fail (error) — The substitution in step 2 uses -4 for f'(2), but the table clearly states f'(2) = 4. This sign error leads to an incorrect calculation.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-26 — The substitution in step 2 uses -4 for f'(2), but the table clearly states f'(2) = 4. This sign error leads to an incorrect calculation.
gpt-oss:20b: pass 2026-10-26
qwen3.6:27b-mlx: fail (error) 2026-10-25 — The substitution in step 2 uses -4 for f'(2), but the table clearly states f'(2) = 4. This sign error leads to the incorrect numerator (-4 + 20 = 16) instead of the correct one (4 + 20 = 24).
gpt-oss:20b: pass 2026-10-25
qwen3.6:27b-mlx: fail (error) 2026-10-25 — The solution substitutes f'(2) = -4, but the table explicitly states f'(2) = 4. This is a transcription error that leads to an incorrect calculation.
gpt-oss:20b: pass 2026-10-25

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.