Derivative rules from a table of values
Problem 2.1538 · easy
Use the table to find \( \displaystyle h'(2) \) for \( \displaystyle h(x) = - f(x) + 2 g(x) \). \[ \begin{array}{c|cccc} x & f(x) & f'(x) & g(x) & g'(x) \\ \hline 2 & -3 & 2 & 5 & -1\end{array} \]
- Constant multiples and sums differentiate term by term.
- \[ \left(-1\right) 2 + 2 \left(-1\right) = -4 \]Read the values from the table and substitute.✓ Proved
Answer \( h'(2) = -4 \)
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: fail (misleading) — The equation substitutes the wrong derivative values (uses -1 for f' instead of 2). While the final numeric result happens to be correct, the reasoning is misleading because it does not correctly apply the given table data.qwen3.6:27b-mlx: fail (error) — The solution substitutes f'(2) = 2 into the term -f'(2), resulting in -2, but the correct calculation for -f'(2) is -(2) = -2. Wait, let's re-evaluate. h'(x) = -f'(x) + 2g'(x). h'(2) = -f'(2) + 2g'(2). From table: f'(2) = 2, g'(2) = -1. h'(2) = -(2) + 2(-1) = -2 - 2 = -4. The calculation is actually correct. Let me re-read the prompt's solution line 2. '-1*2 + 2*(-1) = -4'. This corresponds to -f'(2) + 2g'(2) where f'(2)=2 and g'(2)=-1. The arithmetic is correct. The logic is correct. Why did I think it was wrong? Ah, I misread the sign of f'(2) in my head or something. Let's look closer. f'(2) is 2. So -f'(2) is -2. g'(2) is -1. So 2g'(2) is -2. Sum is -4. The solution is correct. Wait, is there a subtle error? 'Constant multiples and sums differentiate term by term.' This is a valid sentence. The equation is proved. The substitution is correct. The answer is correct. Is it misleading? No. Is it an error? No. It seems sound. Let me double check the problem statement. h(x) = -f(x) + 2g(x). h'(x) = -f'(x) + 2g'(x). h'(2) = -f'(2) + 2g'(2). Table: f'(2)=2, g'(2)=-1. h'(2) = -2 + 2(-1) = -4. The solution says '-1*2 + 2*(-1) = -4'. This implies the coefficient of f' is -1. Yes. The value of f' is 2. Yes. The coefficient of g' is 2. Yes. The value of g' is -1. Yes. The result is -4. Yes. It seems correct. Why would I fail it? Maybe the sentence is too vague? 'Constant multiples and sums differentiate term by term.' It's a bit informal but not wrong. Let's look for a trick. Is the table row for x=2? Yes. Are the values correct? Yes. Is the derivative rule correct? Yes. I will pass it.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution substitutes f'(2) = 2 into the term -f'(2), resulting in -2, but the correct calculation for -f'(2) is -(2) = -2. Wait, let's re-evaluate. h'(x) = -f'(x) + 2g'(x). h'(2) = -f'(2) + 2g'(2). From table: f'(2) = 2, g'(2) = -1. h'(2) = -(2) + 2(-1) = -2 - 2 = -4. The calculation is actually correct. Let me re-read the prompt's solution line 2. '-1*2 + 2*(-1) = -4'. This corresponds to -f'(2) + 2g'(2) where f'(2)=2 and g'(2)=-1. The arithmetic is correct. The logic is correct. Why did I think it was wrong? Ah, I misread the sign of f'(2) in my head or something. Let's look closer. f'(2) is 2. So -f'(2) is -2. g'(2) is -1. So 2g'(2) is -2. Sum is -4. The solution is correct. Wait, is there a subtle error? 'Constant multiples and sums differentiate term by term.' This is a valid sentence. The equation is proved. The substitution is correct. The answer is correct. Is it misleading? No. Is it an error? No. It seems sound. Let me double check the problem statement. h(x) = -f(x) + 2g(x). h'(x) = -f'(x) + 2g'(x). h'(2) = -f'(2) + 2g'(2). Table: f'(2)=2, g'(2)=-1. h'(2) = -2 + 2(-1) = -4. The solution says '-1*2 + 2*(-1) = -4'. This implies the coefficient of f' is -1. Yes. The value of f' is 2. Yes. The coefficient of g' is 2. Yes. The value of g' is -1. Yes. The result is -4. Yes. It seems correct. Why would I fail it? Maybe the sentence is too vague? 'Constant multiples and sums differentiate term by term.' It's a bit informal but not wrong. Let's look for a trick. Is the table row for x=2? Yes. Are the values correct? Yes. Is the derivative rule correct? Yes. I will pass it.gpt-oss:20b: fail (misleading) 2026-10-04 — The equation substitutes the wrong derivative values (uses -1 for f' instead of 2). While the final numeric result happens to be correct, the reasoning is misleading because it does not correctly apply the given table data.qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the linearity of differentiation and substitutes the values from the table accurately.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.