Derivative rules from a table of values
Problem 2.1642 · easy
Use the table to find \( \displaystyle h'(3) \) for \( \displaystyle h(x) = \frac{f(x)}{g(x)} \). \[ \begin{array}{c|cccc} x & f(x) & f'(x) & g(x) & g'(x) \\ \hline 3 & 1 & -1 & 5 & 4\end{array} \]
- Quotient rule: h′ = (f′g − fg′)/g².Reviewed
- \[ - \frac{9}{25} \]Read the values from the table and substitute.✓ Proved
Answer \( h'(3) = - \frac{9}{25} \)
✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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: passqwen3.6:27b-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 2026-10-05 — The solution correctly applies the quotient rule and substitutes the values from the table to arrive at the correct result.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (style) 2026-10-05 — The solution lacks the explicit substitution step showing how the values from the table are plugged into the quotient rule formula, making the derivation opaque.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.