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Derivative rules from a table of values

Problem 2.1959 · easy

Use the table to find \( \displaystyle h'(1) \) for \( \displaystyle h(x) = x f(x) + \left[g(x)\right]^{2} \). \[ \begin{array}{c|cccc} x & f(x) & f'(x) & g(x) & g'(x) \\ \hline 1 & -1 & 2 & 2 & -2\end{array} \]

Product rule on x·f(x), chain rule on g(x)².
Reviewed
\[ 2 \cdot 2 \left(-2\right) - 1 + 1 \cdot 2 = -7 \]
Read the values from the table and substitute.✓ Proved

Answer \( h'(1) = -7 \)

Lines: 1 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the product rule and chain rule to derive h'(x) = f(x) + x f'(x) + 2g(x)g'(x), and the substitution of values from the table is arithmetically correct.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly applies the product rule and chain rule to derive h'(x) = f(x) + x f'(x) + 2g(x)g'(x), and the substitution of values from the table is arithmetically correct.
gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution fails to explicitly state the derivative formula h'(x) = f(x) + x f'(x) + 2 g(x) g'(x) before substituting values. While the arithmetic in line 2 is correct, omitting the symbolic differentiation step makes it unclear how the terms were derived and risks confusing the student about the application of the product and chain rules.

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.