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Derivative rules from a table of values

Problem 2.2507 · easy

Use the table to find \( \displaystyle h'(3) \) for \( \displaystyle h(x) = f{\left(g(x) \right)} \). \[ \begin{array}{c|cccc} x & f(x) & f'(x) & g(x) & g'(x) \\ \hline 3 & -3 & -3 & 4 & -4 \\ 4 & 6 & -4 & \cdot & \cdot\end{array} \]

Chain rule: h′(a) = f′(g(a))·g′(a).
\[ \left(-4\right) \left(-4\right) = 16 \]
Read the values from the table and substitute.✓ Proved

Answer \( h'(3) = 16 \)

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 explicitly evaluate the inner function g(3) to determine the correct input for f'. While the final arithmetic is correct, skipping the step of identifying that f' must be evaluated at g(3)=4 makes the logic incomplete and hides the crucial dependency on the table's second row.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-26 — The solution fails to explicitly evaluate the inner function g(3) to determine the correct input for f'. While the final arithmetic is correct, skipping the step of identifying that f' must be evaluated at g(3)=4 makes the logic incomplete and hides the crucial dependency on the table's second row.
gpt-oss:20b: pass 2026-10-26
qwen3.6:27b-mlx: fail (misleading) 2026-10-25 — The solution skips the crucial step of evaluating g(3) to determine the input for f'. It jumps directly to multiplying f' and g' values without showing that f'(g(3)) becomes f'(4), which is necessary to justify using the row for x=4.
gpt-oss:20b: pass 2026-10-25
qwen3.6:27b-mlx: fail (error) 2026-10-25 — The solution fails to explicitly evaluate the inner function g(3) to determine the correct input for f'. While the final numerical substitution is correct, omitting the step h'(3) = f'(g(3)) * g'(3) = f'(4) * (-4) obscures the critical logic of the chain rule, which requires finding the value of the inner function first.
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.