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Derivative rules from a table of values

Problem 2.2500 · easy

Use the table to find \( \displaystyle h'(1) \) for \( \displaystyle h(x) = x f(x) + \left[g(x)\right]^{2} \). \[ \begin{array}{c|cccc} x & f(x) & f'(x) & g(x) & g'(x) \\ \hline 1 & 5 & 1 & 1 & 1\end{array} \]

Product rule on x·f(x), chain rule on g(x)².
\[ 1 \cdot 1 + 2 \cdot 1 \cdot 1 + 5 = 8 \]
Read the values from the table and substitute.✓ Proved

Answer \( h'(1) = 8 \)

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 explicitly state the derived formula for h'(x) before substituting values. It jumps from naming the rules to a numerical expression, omitting the crucial step of showing that h'(x) = f(x) + x f'(x) + 2 g(x) g'(x). This makes the origin of the terms '1*1', '2*1*1', and '5' opaque and risks teaching students to substitute without understanding the structure of the derivative.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-26 — The solution fails to explicitly state the derived formula for h'(x) before substituting values. It jumps from naming the rules to a numerical expression, omitting the crucial step of showing that h'(x) = f(x) + x f'(x) + 2 g(x) g'(x). This makes the origin of the terms '1*1', '2*1*1', and '5' opaque and risks teaching students to substitute without understanding the structure of the derivative.
gpt-oss:20b: pass 2026-10-26
qwen3.6:27b-mlx: fail (error) 2026-10-25 — The solution fails to explicitly state the derived formula for h'(x) before substituting values. While the arithmetic in line 2 is correct, line 1 only lists rules without showing how they combine into h'(x) = f(x) + x f'(x) + 2g(x)g'(x), making the jump to the numerical expression in line 2 opaque and skipping the crucial modeling step.
gpt-oss:20b: pass 2026-10-25
qwen3.6:27b-mlx: fail (error) 2026-10-25 — The solution fails to explicitly state the derived formula for h'(x) before substitution, making the arithmetic in step 2 appear unmotivated and obscuring the application of the product and chain rules. Specifically, it skips showing that h'(x) = f(x) + x f'(x) + 2g(x)g'(x), which is necessary to verify that the terms 1*1, 2*1*1, and 5 correspond correctly to the components of the derivative.
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.