Derivative rules from a table of values
Problem 2.2097 · easy
Use the table to find \( \displaystyle h'(1) \) for \( \displaystyle h(x) = x f(x) + \left[g(x)\right]^{2} \). \[ \begin{array}{c|cccc} x & f(x) & f'(x) & g(x) & g'(x) \\ \hline 1 & -5 & 1 & 1 & -2\end{array} \]
- Product rule on x·f(x), chain rule on g(x)².Reviewed
- \[ -5 + 2 \cdot 1 \left(-2\right) + 1 \cdot 1 = -8 \]Read the values from the table and substitute.✓ Proved
Answer \( h'(1) = -8 \)
Lines: 1 proved, 1 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: passqwen3.6:27b-mlx: pass — The solution correctly identifies the differentiation rules needed (product rule for the first term, chain rule for the second) and correctly substitutes the values from the table to arrive at the stated answer.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly identifies the differentiation rules needed (product rule for the first term, chain rule for the second) and correctly substitutes the values from the table to arrive at the stated answer.gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: fail (error) 2026-10-11 — The derivative calculation in line 2 is incorrect. The product rule for x*f(x) yields 1*f(1) + 1*f'(1) = -5 + 1 = -4. The chain rule for [g(x)]^2 yields 2*g(1)*g'(1) = 2*1*(-2) = -4. The sum is -8, but the equation written as '-5 + 2*1*(-2) + 1*1' evaluates to -5 - 4 + 1 = -8, which coincidentally gives the right answer but represents the wrong terms (it mixes f(1) with the derivative term incorrectly or implies f'(1) is 1 and f(1) is -5 but adds them in a confusing way). Actually, looking closer: h'(x) = f(x) + x f'(x) + 2 g(x) g'(x). At x=1: -5 + 1(1) + 2(1)(-2) = -5 + 1 - 4 = -8. The equation in line 2 is '-5 + 2*1*(-2) + 1*1'. This is -5 - 4 + 1 = -8. The terms are f(1), 2g(1)g'(1), and x f'(1). This is actually correct algebraically. Wait, let me re-read the sentence. 'Product rule on x·f(x), chain rule on g(x)².' This is correct. The equation substitutes: f(1)=-5, 2*g(1)*g'(1) = 2*1*-2, and 1*f'(1) = 1*1. Sum = -5 -4 + 1 = -8. The solution is actually correct. Why did I think it was wrong? Let me re-evaluate. h'(x) = (x)'f(x) + x(f(x))' + 2g(x)g'(x) = 1*f(x) + x*f'(x) + 2g(x)g'(x). At x=1: 1*(-5) + 1*(1) + 2*(1)*(-2) = -5 + 1 - 4 = -8. The equation in line 2 is '-5 + 2*1*(-2) + 1*1'. This corresponds to f(1) + 2g(1)g'(1) + x f'(1). This is correct. The verdict should be pass.gpt-oss:20b: pass 2026-10-11
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 2026-10-11 with SymPy 1.14.0.