Derivative rules from a table of values
Problem 2.1804 · easy
Use the table to find \( \displaystyle h'(1) \) for \( \displaystyle h(x) = \frac{f(x)}{g(x)} \). \[ \begin{array}{c|cccc} x & f(x) & f'(x) & g(x) & g'(x) \\ \hline 1 & 5 & 2 & -1 & 1\end{array} \]
- Quotient rule: h′ = (f′g − fg′)/g².
- \[ -7 \]Read the values from the table and substitute.✓ Proved
Answer \( h'(1) = -7 \)
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: fail (misleading) — The solution skips the actual substitution step; it merely states "-7 = -7" without showing how the quotient rule yields that value, which could mislead a student into thinking the computation is trivial or already proven.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: fail (misleading) 2026-10-07 — The solution skips the actual substitution step; it merely states "-7 = -7" without showing how the quotient rule yields that value, which could mislead a student into thinking the computation is trivial or already proven.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: pass 2026-10-07 — The solution correctly applies the quotient rule and substitutes the values from the table to arrive at the correct result.gpt-oss:20b: pass 2026-10-07
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.