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Derivative rules from a table of values

Problem 2.2501 · easy

Use the table to find \( \displaystyle h'(2) \) for \( \displaystyle h(x) = - 4 f(x) - 2 g(x) \). \[ \begin{array}{c|cccc} x & f(x) & f'(x) & g(x) & g'(x) \\ \hline 2 & 1 & 2 & 1 & 2\end{array} \]

Constant multiples and sums differentiate term by term.
\[ \left(-4\right) 2 - 2 \cdot 2 = -12 \]
Read the values from the table and substitute.✓ Proved

Answer \( h'(2) = -12 \)

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. How we verify

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: pass
qwen3.6:27b-mlx: fail (misleading) — The solution omits the crucial step of explicitly stating the derivative formula h'(x) = -4f'(x) - 2g'(x). It jumps from the rule to the arithmetic, failing to show that the values 2 and 2 come from f'(2) and g'(2) respectively, not f(2) and g(2). This obscures the application of the differentiation rules.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (misleading) 2026-10-26 — The solution omits the crucial step of explicitly stating the derivative formula h'(x) = -4f'(x) - 2g'(x). It jumps from the rule to the arithmetic, failing to show that the values 2 and 2 come from f'(2) and g'(2) respectively, not f(2) and g(2). This obscures the application of the differentiation rules.
gpt-oss:20b: pass 2026-10-26
qwen3.6:27b-mlx: fail (error) 2026-10-25 — The solution fails to explicitly state the differentiation rule h'(x) = -4f'(x) - 2g'(x) before substituting values. While the arithmetic is correct, omitting the symbolic derivative step makes the logic opaque and skips the crucial application of the linearity of differentiation.
gpt-oss:20b: pass 2026-10-25
qwen3.6:27b-mlx: pass 2026-10-25 — The solution correctly applies the linearity of the derivative operator and substitutes the values from the table accurately.
gpt-oss:20b: pass 2026-10-25

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 2026-10-26 with SymPy 1.14.0.