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Derivative rules from a table of values

Problem 2.2508 · easy

Use the table to find \( \displaystyle h'(0) \) for \( \displaystyle h(x) = f{\left(g(x) \right)} \). \[ \begin{array}{c|cccc} x & f(x) & f'(x) & g(x) & g'(x) \\ \hline 0 & 5 & -4 & -2 & 2 \\ -2 & -5 & -4 & \cdot & \cdot\end{array} \]

Chain rule: h′(a) = f′(g(a))·g′(a).
\[ \left(-4\right) 2 = -8 \]
Read the values from the table and substitute.✓ Proved

Answer \( h'(0) = -8 \)

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 (error) — The solution fails to evaluate the inner function g(0) to determine the correct argument for f'. It incorrectly assumes f' is evaluated at 0, whereas the chain rule requires f'(g(0)) = f'(-2). Although the table happens to list f'(-2) = -4, the reasoning is flawed because it skips the crucial step of identifying the correct input value.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-26 — The solution fails to evaluate the inner function g(0) to determine the correct argument for f'. It incorrectly assumes f' is evaluated at 0, whereas the chain rule requires f'(g(0)) = f'(-2). Although the table happens to list f'(-2) = -4, the reasoning is flawed because it skips the crucial step of identifying the correct input value.
gpt-oss:20b: pass 2026-10-26
qwen3.6:27b-mlx: fail (error) 2026-10-25 — The solution fails to explicitly evaluate the inner function g(0) to determine the correct argument for f'. While the final numerical substitution is correct, omitting the step f'(g(0)) = f'(-2) hides the critical logic of the chain rule, which requires looking up f' at the value of g(0), not just plugging in numbers blindly.
gpt-oss:20b: pass 2026-10-25
qwen3.6:27b-mlx: fail (error) 2026-10-25 — The solution substitutes f'(0) instead of f'(g(0)). Since g(0) = -2, the correct value is f'(-2) = -4, but the reasoning implies using the row for x=0 for f', which is a logical error in applying the chain rule structure, even though the numerical coincidence of f'(0) = f'(-2) yields the correct answer.
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.