Derivative rules from a table of values
Problem 2.1956 · easy
Use the table to find \( \displaystyle h'(0) \) for \( \displaystyle h(x) = x f(x) + \left[g(x)\right]^{2} \). \[ \begin{array}{c|cccc} x & f(x) & f'(x) & g(x) & g'(x) \\ \hline 0 & 4 & -4 & -1 & 1\end{array} \]
- Product rule on x·f(x), chain rule on g(x)².
- \[ 2 \left(-1\right) 1 + 0 \left(-4\right) + 4 = 2 \]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 substituting values, making the arithmetic in step 2 appear unmotivated and obscuring the application of the product and chain rules. Specifically, it skips showing that h'(x) = f(x) + x f'(x) + 2g(x)g'(x), which is necessary to verify the correct substitution of terms.
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 substituting values, making the arithmetic in step 2 appear unmotivated and obscuring the application of the product and chain rules. Specifically, it skips showing that h'(x) = f(x) + x f'(x) + 2g(x)g'(x), which is necessary to verify the correct substitution of terms.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, making it unclear how the terms correspond to the product and chain rules. Specifically, it omits the term f(x) from the product rule derivative of x*f(x) in the explanation, although the numerical substitution (4) happens to be correct, the logical gap is significant.
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.