Derivative rules from a table of values
Problem 2.1541 · easy
Use the table to find \( \displaystyle h'(2) \) for \( \displaystyle h(x) = x f(x) + \left[g(x)\right]^{2} \). \[ \begin{array}{c|cccc} x & f(x) & f'(x) & g(x) & g'(x) \\ \hline 2 & 4 & 1 & 1 & -2\end{array} \]
- Product rule on x·f(x), chain rule on g(x)².
- \[ 2 \cdot 1 \left(-2\right) + 2 \cdot 1 + 4 = 2 \]Read the values from the table and substitute.✓ Proved
Answer \( h'(2) = 2 \)
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 derivative calculation is incorrect. The product rule for x*f(x) yields f(x) + x*f'(x), which evaluates to 4 + 2(1) = 6, not 2. The chain rule for g(x)^2 yields 2*g(x)*g'(x), which evaluates to 2(1)(-2) = -4. The sum is 6 + (-4) = 2, but the intermediate equation provided (2*1*(-2) + 2*1 + 4 = 2) incorrectly combines terms and misapplies the product rule structure, leading to a correct final number by coincidence but via flawed logic.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The derivative calculation is incorrect. The product rule for x*f(x) yields f(x) + x*f'(x), which evaluates to 4 + 2(1) = 6, not 2. The chain rule for g(x)^2 yields 2*g(x)*g'(x), which evaluates to 2(1)(-2) = -4. The sum is 6 + (-4) = 2, but the intermediate equation provided (2*1*(-2) + 2*1 + 4 = 2) incorrectly combines terms and misapplies the product rule structure, leading to a correct final number by coincidence but via flawed logic.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (error) 2026-10-04 — The derivative calculation is incorrect. The term for the derivative of [g(x)]^2 is 2g(x)g'(x), which evaluates to 2(1)(-2) = -4, not 2*1*(-2) treated as a separate additive term without the factor of 2 from the chain rule being applied correctly to the whole expression, or rather, the substitution in line 2 is arithmetically inconsistent with the stated rules. Specifically, h'(x) = f(x) + x f'(x) + 2g(x)g'(x). At x=2, this is 4 + 2(1) + 2(1)(-2) = 4 + 2 - 4 = 2. Wait, the final answer 2 is actually correct. Let me re-read the solution line 2 carefully. Line 2 says: '2*1*(-2) + 2*1 + 4 = 2'. Let's parse this. 2*1*(-2) is -4. 2*1 is 2. 4 is 4. Sum is 2. This matches the correct calculation. However, the sentence in step 1 says 'Product rule on x·f(x), chain rule on g(x)²'. This is correct. The equation in step 2 is marked 'unchecked' but the prompt says 'Each line of the solution is either an EQUATION, which a computer algebra system has already checked... or a SENTENCE'. Wait, the prompt says '2. [equation, unchecked]'. This is a contradiction in the prompt description vs standard behavior, but I must review the content. The equation itself: 2*1*(-2) + 2*1 + 4 = 2. This corresponds to 2*g(2)*g'(2) + 2*f'(2) + f(2)? No. h'(x) = 1*f(x) + x*f'(x) + 2*g(x)*g'(x). At x=2: f(2)=4, f'(2)=1, g(2)=1, g'(2)=-2. h'(2) = 4 + 2(1) + 2(1)(-2) = 4 + 2 - 4 = 2. The equation in line 2 is: 2*1*(-2) + 2*1 + 4. This is -4 + 2 + 4 = 2. The terms are: 2*g(2)*g'(2) is -4. x*f'(2) is 2*1=2. f(2) is 4. The order is swapped but the values are correct. The sentence in step 1 is correct. The equation in step 2 is correct. Why would it fail? Let me look closer. Is there a subtle error? '2*1*(-2)' is the chain rule part. '2*1' is the product rule part (x f'(x)). '4' is the product rule part (f(x)). The calculation is correct. The answer is correct. The setup is correct. Is it misleading? No. Is it style? Maybe the lack of explicit formula before substitution is poor style, but not an error. Wait, look at the prompt again. '2. [equation, unchecked]'. If the CAS checked it, it's fine. But the prompt says 'do not re-check the algebra of a line marked proved'. Here it is marked 'unchecked'. This might imply I should check it? Or is it a trick? The prompt says 'Each line ... is either an EQUATION ... (its result is shown; do not re-check ...) OR a SENTENCE'. Line 2 is labeled '[equation, unchecked]'. This label is weird. Usually, if it's an equation, the CAS checks it. If it's unchecked, maybe it's not a valid equation line? Or maybe the user made a typo in the label? Regardless, the math is correct. Let me double check the derivative. h(x) = x f(x) + g(x)^2. h'(x) = f(x) + x f'(x) + 2 g(x) g'(x). h'(2) = 4 + 2(1) + 2(1)(-2) = 4 + 2 - 4 = 2. The solution gets 2. The steps are logically sound. I will pass it.gpt-oss:20b: pass 2026-10-04
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 2026-10-04 with SymPy 1.14.0.