Derivative rules from a table of values
Problem 2.1957 · easy
Use the table to find \( \displaystyle h'(3) \) for \( \displaystyle h(x) = f{\left(g(x) \right)} \). \[ \begin{array}{c|cccc} x & f(x) & f'(x) & g(x) & g'(x) \\ \hline 3 & 1 & 2 & 2 & -1 \\ 2 & 5 & 1 & \cdot & \cdot\end{array} \]
- Chain rule: h′(a) = f′(g(a))·g′(a).
- \[ 1 \left(-1\right) = -1 \]Read the values from the table and substitute.✓ Proved
Answer \( h'(3) = -1 \)
Lines: 1 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see 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: fail (error) — The solution incorrectly uses f′(g(3)) = f′(3). The table gives f′ only at x=3, not at x=2, so h′(3) cannot be computed from the provided data. The conclusion h′(3)=−1 is unsupported.qwen3.6:27b-mlx: fail (error) — The solution substitutes f'(g(3)) with 1, but the table shows f'(2) = 1. While the final numerical answer is correct, the sentence implies a direct substitution without explicitly evaluating the inner function g(3) = 2 first, which is a critical step in applying the chain rule. A student might incorrectly assume f'(g(3)) is read directly from the row x=3, rather than looking up f' at the value g(3)=2.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-09 — The solution incorrectly uses f′(g(3)) = f′(3). The table gives f′ only at x=3, not at x=2, so h′(3) cannot be computed from the provided data. The conclusion h′(3)=−1 is unsupported.qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution substitutes f'(g(3)) with 1, but the table shows f'(2) = 1. While the final numerical answer is correct, the sentence implies a direct substitution without explicitly evaluating the inner function g(3) = 2 first, which is a critical step in applying the chain rule. A student might incorrectly assume f'(g(3)) is read directly from the row x=3, rather than looking up f' at the value g(3)=2.gpt-oss:20b: fail (error) 2026-10-09 — The chain rule requires f′ evaluated at g(3), i.e. f′(2), but the table only provides f′(3). Without f′(2) the computation is invalid.qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution substitutes f'(3) instead of f'(g(3)). Since g(3) = 2, the correct value is f'(2) = 1, but the logic implies using the row for x=3 for f', which is incorrect application of the chain rule.
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.