Derivative rules from a table of values
Problem 2.1802 · easy
Use the table to find \( \displaystyle h'(0) \) for \( \displaystyle h(x) = x f(x) + \left[g(x)\right]^{2} \). \[ \begin{array}{c|cccc} x & f(x) & f'(x) & g(x) & g'(x) \\ \hline 0 & -4 & -3 & 1 & 1\end{array} \]
- Product rule on x·f(x), chain rule on g(x)².
- \[ -4 + 0 \left(-3\right) + 2 \cdot 1 \cdot 1 = -2 \]Read the values from the table and substitute.✓ Proved
Answer \( h'(0) = -2 \)
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, making the jump from the rules to the numerical expression opaque. Specifically, it omits showing that h'(x) = f(x) + x f'(x) + 2g(x)g'(x), which is necessary to verify that the terms -4, 0*(-3), and 2*1*1 correspond correctly to the product and chain rule applications.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to explicitly state the derived formula for h'(x) before substitution, making the jump from the rules to the numerical expression opaque. Specifically, it omits showing that h'(x) = f(x) + x f'(x) + 2g(x)g'(x), which is necessary to verify that the terms -4, 0*(-3), and 2*1*1 correspond correctly to the product and chain rule applications.qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to explicitly state the derived formula for h'(x) before substituting values. While the arithmetic is correct, omitting the symbolic differentiation step (h'(x) = f(x) + x f'(x) + 2 g(x) g'(x)) makes it impossible to verify that the correct rules were applied, rather than just guessing the numbers.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.