Derivative rules from a table of values
Problem 2.2105 · easy
Use the table to find \( \displaystyle h'(2) \) for \( \displaystyle h(x) = f{\left(g(x) \right)} \). \[ \begin{array}{c|cccc} x & f(x) & f'(x) & g(x) & g'(x) \\ \hline 2 & -5 & -1 & -1 & 5 \\ -1 & -3 & 5 & \cdot & \cdot\end{array} \]
- Chain rule: h′(a) = f′(g(a))·g′(a).
- \[ 5 \cdot 5 = 25 \]Read the values from the table and substitute.✓ Proved
Answer \( h'(2) = 25 \)
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: passqwen3.6:27b-mlx: fail (error) — The solution incorrectly substitutes g(2) = -1 with 2 when evaluating f'. It calculates f'(2) * g'(2) instead of f'(g(2)) * g'(2), which should be f'(-1) * g'(2) = 5 * 5 = 25. While the final numerical answer happens to be correct due to f'(2) coincidentally equaling f'(-1) in this specific table, the reasoning is fundamentally flawed and teaches the wrong application of the chain rule.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution incorrectly substitutes g(2) = -1 with 2 when evaluating f'. It calculates f'(2) * g'(2) instead of f'(g(2)) * g'(2), which should be f'(-1) * g'(2) = 5 * 5 = 25. While the final numerical answer happens to be correct due to f'(2) coincidentally equaling f'(-1) in this specific table, the reasoning is fundamentally flawed and teaches the wrong application of the chain rule.gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution incorrectly substitutes g(2) = -1 into f' as if it were 2, using f'(2) = -1 instead of f'(g(2)) = f'(-1) = 5. The correct calculation is 5 * 5 = 25, but the reasoning shown (implied by just writing 5*5 without showing the lookup of f'(-1)) is ambiguous and likely hides the error of using the wrong row for f'. Wait, looking at the table: f'(-1) is 5. g'(2) is 5. So 5*5=25 is the correct numerical answer. However, the sentence 'Read the values from the table and substitute' is too vague. It doesn't explicitly state that we need f'(g(2)) = f'(-1). A student might think we just multiplied the two 5s in the row for x=2. This is misleading because it obscures the crucial step of evaluating the inner function first. Let's look closer. The solution says '5*5 = 25'. Where do the 5s come from? g'(2)=5. f'(g(2)) = f'(-1) = 5. The solution does not show the lookup of f'(-1). It just says 'Read the values... and substitute'. This is a 'misleading' severity because it fails to demonstrate the chain rule application correctly, potentially teaching that you just multiply numbers from the same row or random 5s. Actually, is it an error? The final answer is correct. But the explanation is insufficient to verify the method. Is it 'error' or 'misleading'? If a student follows this, they might not know to look up f'(-1). I will call it 'misleading' because the logic is skipped, not because the math is wrong, but the prompt asks to find what is wrong. The 'wrong' thing is the lack of explicit substitution showing g(2)=-1. However, often 'error' is reserved for wrong answers or wrong theorems. Let's look at the options. 'error' (a step or the setup is wrong). The step '5*5=25' is arithmetically correct. The setup is correct. The missing link is the evaluation of f'(g(2)). This is a gap in reasoning. Is it an error? No. Is it misleading? Yes. A student might think h'(2) = f'(2)*g'(2) = -1*5 = -5 is wrong, but if they see 5*5, they might guess where the 5s came from. The bigger issue is that the solution doesn't explicitly state f'(g(2)) = f'(-1) = 5. It just jumps to 5*5. This is a classic 'skip step' error in pedagogy. I will mark it as 'misleading' because it doesn't teach the correct procedure clearly.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.