Derivative rules from a table of values
Problem 2.2096 · easy
Use the table to find \( \displaystyle h'(2) \) for \( \displaystyle h(x) = f{\left(g(x) \right)} \). \[ \begin{array}{c|cccc} x & f(x) & f'(x) & g(x) & g'(x) \\ \hline 2 & 2 & 2 & -2 & -4 \\ -2 & -4 & 4 & \cdot & \cdot\end{array} \]
- Chain rule: h′(a) = f′(g(a))·g′(a).Reviewed
- \[ 4 \left(-4\right) = -16 \]Read the values from the table and substitute.✓ Proved
Answer \( h'(2) = -16 \)
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 applies the chain rule and substitutes the appropriate values from the table (f'(-2) = 4 and g'(2) = -4) to arrive at the correct result.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly applies the chain rule and substitutes the appropriate values from the table (f'(-2) = 4 and g'(2) = -4) to arrive at the correct result.gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution incorrectly substitutes f'(g(2)) as f'(-2) = 4, but the table does not provide g'(-2), making it impossible to verify if g(2)=-2 leads to a valid evaluation or if the student simply misread the table. More critically, the chain rule requires evaluating f' at g(2). The table gives g(2) = -2. So we need f'(-2). The table gives f'(-2) = 4. And g'(2) = -4. So 4 * -4 = -16. Wait, let me re-read carefully. Problem: h(x) = f(g(x)). h'(x) = f'(g(x)) * g'(x). h'(2) = f'(g(2)) * g'(2). From table: g(2) = -2. g'(2) = -4. So h'(2) = f'(-2) * (-4). From table: f'(-2) = 4. So h'(2) = 4 * (-4) = -16. The solution says: '4*(-4) = -16 -- Read the values from the table and substitute.' It does not explicitly state that f'(g(2)) becomes f'(-2). It just jumps to the numbers. Is this an error? The prompt asks to review sentences. Sentence 1 is correct. Sentence 2 is an equation marked 'unchecked' but the instruction says 'equation... checked... do not re-check'. However, the text '-- Read the values...' is part of the line. Let's look closer. Is there a trap? x=2: f(2)=2, f'(2)=2, g(2)=-2, g'(2)=-4. x=-2: f(-2)=-4, f'(-2)=4. Calculation: f'(g(2)) = f'(-2) = 4. g'(2) = -4. Product = -16. The answer is correct. The steps are logically sound, though brief. Is there anything wrong? Sentence 1: Chain rule formula is correct. Sentence 2: The substitution is correct. Why would it fail? Maybe the user thinks f'(g(2)) should be f'(2)? No, g(2) is -2. Maybe the user thinks we need g'(-2)? No, we need g'(2). The solution is actually correct. Wait, look at the table again. Row 1: x=2. g(2) = -2. g'(2) = -4. Row 2: x=-2. f'(-2) = 4. The solution uses 4 and -4. 4 comes from f'(-2). -4 comes from g'(2). The text says 'Read the values from the table and substitute.' It doesn't explicitly explain the composition step g(2)=-2. For a calculus student, skipping the intermediate step of identifying g(2) might be considered 'misleading' or poor style, but is it an error? Let's check the constraints. 'Check that... each conclusion follows from the lines before it'. Line 1: h'(a) = f'(g(a))g'(a). Line 2: 4*(-4) = -16. Does Line 2 follow from Line 1? Only if you implicitly know g(2)=-2 and f'(-2)=4. The sentence 'Read the values...' asserts this. Is it possible the solution is wrong? What if the student confused f'(2) and f'(-2)? f'(2) = 2. If they used that, they'd get 2*-4 = -8. They used 4. 4 is f'(-2). This is the correct value for f'(g(2)). So the math is right. Is there a subtle error? 'h'(2) = -16'. Let's look at the 'unchecked' tag. The prompt says 'equation... checked... do not re-check'. I will mark it as pass. The logic is sound, even if brief.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.