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Derivative rules from a table of values

Problem 2.1536 · easy

Use the table to find \( \displaystyle h'(0) \) for \( \displaystyle h(x) = \frac{f(x)}{g(x)} \). \[ \begin{array}{c|cccc} x & f(x) & f'(x) & g(x) & g'(x) \\ \hline 0 & 3 & 4 & 4 & 2\end{array} \]

Quotient rule: h′ = (f′g − fg′)/g².
Reviewed
\[ \frac{5}{8} \]
Read the values from the table and substitute.✓ Proved

Answer \( h'(0) = \frac{5}{8} \)

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)

qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the quotient rule and substitutes the values from the table to arrive at the correct result.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: fail (misleading) 2026-10-04 — The solution skips the explicit substitution step, making it impossible to verify that the values were read correctly from the table. A student should see h'(0) = (4*4 - 3*2)/4^2 = 10/16 = 5/8 to learn the process.
gpt-oss:20b: pass 2026-10-04

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 2026-10-04 with SymPy 1.14.0.