Derivative rules from a table of values
Problem 2.1649 · easy
Use the table to find \( \displaystyle h'(3) \) for \( \displaystyle h(x) = \frac{f(x)}{g(x)} \). \[ \begin{array}{c|cccc} x & f(x) & f'(x) & g(x) & g'(x) \\ \hline 3 & 6 & 5 & 5 & -4\end{array} \]
- Quotient rule: h′ = (f′g − fg′)/g².
- \[ \frac{49}{25} \]Read the values from the table and substitute.✓ Proved
Answer \( h'(3) = \frac{49}{25} \)
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.
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: passqwen3.6:27b-mlx: fail (error) — The solution claims to substitute values from the table but provides no evidence of the substitution step, making it impossible to verify the arithmetic or the application of the quotient rule. The equation line is a tautology that does not show the work.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution claims to substitute values from the table but provides no evidence of the substitution step, making it impossible to verify the arithmetic or the application of the quotient rule. The equation line is a tautology that does not show the work.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution claims to substitute values from the table but provides no evidence of the substitution step. It jumps directly to the final answer, failing to show the calculation (5*5 - 6*(-4))/5^2, which makes it impossible to verify if the student actually performed the arithmetic correctly or just guessed the answer.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.