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Derivative rules from a table of values

Problem 2.1645 · easy

Use the table to find \( \displaystyle h'(0) \) for \( \displaystyle h(x) = f{\left(g(x) \right)} \). \[ \begin{array}{c|cccc} x & f(x) & f'(x) & g(x) & g'(x) \\ \hline 0 & 4 & 4 & -1 & -1 \\ -1 & -2 & -1 & \cdot & \cdot\end{array} \]

Chain rule: h′(a) = f′(g(a))·g′(a).
\[ \left(-1\right) \left(-1\right) = 1 \]
Read the values from the table and substitute.✓ Proved

Answer \( h'(0) = 1 \)

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: pass
qwen3.6:27b-mlx: fail (error) — The solution substitutes g'(0) = -1 and f'(-1) = -1, but fails to explicitly state that f'(g(0)) evaluates to f'(-1). While the arithmetic is correct, the logical step of identifying the inner function's value as the input for the outer derivative is missing, which is the core of the chain rule application.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution substitutes g'(0) = -1 and f'(-1) = -1, but fails to explicitly state that f'(g(0)) evaluates to f'(-1). While the arithmetic is correct, the logical step of identifying the inner function's value as the input for the outer derivative is missing, which is the core of the chain rule application.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution skips the crucial step of evaluating the inner function g(0) to determine the argument for f'. It substitutes g'(0) directly into f' without showing that f' is evaluated at g(0) = -1, which is the core concept of the chain rule.
gpt-oss:20b: pass 2026-10-05

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 2026-10-05 with SymPy 1.14.0.