Derivative rules from a table of values
Problem 2.1962 · easy
Use the table to find \( \displaystyle h'(1) \) for \( \displaystyle h(x) = x f(x) + \left[g(x)\right]^{2} \). \[ \begin{array}{c|cccc} x & f(x) & f'(x) & g(x) & g'(x) \\ \hline 1 & -4 & -2 & 5 & -1\end{array} \]
- Product rule on x·f(x), chain rule on g(x)².
- \[ 2 \cdot 5 \left(-1\right) - 4 + 1 \left(-2\right) = -16 \]Read the values from the table and substitute.✓ Proved
Answer \( h'(1) = -16 \)
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 fails to explicitly state the derived formula for h'(x) before substitution. While the arithmetic in line 2 is correct, omitting the symbolic differentiation step (h'(x) = f(x) + xf'(x) + 2g(x)g'(x)) makes it impossible to verify that the correct rules were applied, and it obscures the structure of the calculation for the student.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution fails to explicitly state the derived formula for h'(x) before substitution. While the arithmetic in line 2 is correct, omitting the symbolic differentiation step (h'(x) = f(x) + xf'(x) + 2g(x)g'(x)) makes it impossible to verify that the correct rules were applied, and it obscures the structure of the calculation for the student.qwen3.6:27b-mlx: inconclusive 2026-10-09 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The derivative calculation is incorrect. The product rule for x f(x) yields f(x) + x f'(x), which evaluates to -4 + 1(-2) = -6, not -4 + 1(-2) treatgpt-oss:20b: pass 2026-10-09
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 2026-10-09 with SymPy 1.14.0.