Derivative rules from a table of values
Problem 2.1798 · easy
Use the table to find \( \displaystyle h'(2) \) for \( \displaystyle h(x) = - 3 f(x) - 3 g(x) \). \[ \begin{array}{c|cccc} x & f(x) & f'(x) & g(x) & g'(x) \\ \hline 2 & 6 & -3 & -3 & 5\end{array} \]
- Constant multiples and sums differentiate term by term.
- \[ \left(-3\right) 5 - 3 \left(-3\right) = -6 \]Read the values from the table and substitute.✓ Proved
Answer \( h'(2) = -6 \)
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 substitutes g'(2) = 5 into the term -3g(x), but the derivative of -3g(x) is -3g'(x). The calculation -3*5 - 3*(-3) uses g'(2) for the first term (correct for -3g'(x)) but uses g(2) = -3 for the second term (incorrect, should be g'(2) again or the formula is misapplied). Wait, let's re-read carefully. h(x) = -3f(x) - 3g(x). h'(x) = -3f'(x) - 3g'(x). h'(2) = -3f'(2) - 3g'(2). From table: f'(2) = -3, g'(2) = 5. So h'(2) = -3(-3) - 3(5) = 9 - 15 = -6. The equation line says: -3*5 - 3*(-3) = -6. This corresponds to -3*g'(2) - 3*f'(2). The order is swapped compared to the function definition, but the values are swapped too? No. Term 1 in equation: -3*5. 5 is g'(2). So this is -3g'(2). Term 2 in equation: -3*(-3). -3 is f'(2). So this is -3f'(2). The sum is -3g'(2) - 3f'(2). This equals h'(2). The calculation is correct. However, the sentence says 'Constant multiples and sums differentiate term by term.' This is vague but acceptable. The equation line is marked 'proved' by CAS, so I don't check algebra. But does the sentence support the equation? The equation calculates -3*5 - 3*(-3). This is -3g'(2) - 3f'(2). The problem asks for h'(2) where h = -3f - 3g. So h' = -3f' - 3g'. The equation computes -3g' - 3f'. This is the same value. Is there an error? Let's look closer. The equation is `-3*5 - 3*(-3)`. 5 is g'(2). -3 is f'(2). So it is `-3*g'(2) - 3*f'(2)`. This is equal to `h'(2)`. The result is -6. The stated answer is -6. The solution is correct. Why did I think it was wrong? I misread the mapping. Let's re-verify. h'(2) = -3f'(2) - 3g'(2) = -3(-3) - 3(5) = 9 - 15 = -6. The equation line is `-3*5 - 3*(-3) = -15 + 9 = -6`. It uses the values correctly, just in a different order (g term first, then f term). This is mathematically valid. The sentence is generic but true. Verdict: pass.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution substitutes g'(2) = 5 into the term -3g(x), but the derivative of -3g(x) is -3g'(x). The calculation -3*5 - 3*(-3) uses g'(2) for the first term (correct for -3g'(x)) but uses g(2) = -3 for the second term (incorrect, should be g'(2) again or the formula is misapplied). Wait, let's re-read carefully. h(x) = -3f(x) - 3g(x). h'(x) = -3f'(x) - 3g'(x). h'(2) = -3f'(2) - 3g'(2). From table: f'(2) = -3, g'(2) = 5. So h'(2) = -3(-3) - 3(5) = 9 - 15 = -6. The equation line says: -3*5 - 3*(-3) = -6. This corresponds to -3*g'(2) - 3*f'(2). The order is swapped compared to the function definition, but the values are swapped too? No. Term 1 in equation: -3*5. 5 is g'(2). So this is -3g'(2). Term 2 in equation: -3*(-3). -3 is f'(2). So this is -3f'(2). The sum is -3g'(2) - 3f'(2). This equals h'(2). The calculation is correct. However, the sentence says 'Constant multiples and sums differentiate term by term.' This is vague but acceptable. The equation line is marked 'proved' by CAS, so I don't check algebra. But does the sentence support the equation? The equation calculates -3*5 - 3*(-3). This is -3g'(2) - 3f'(2). The problem asks for h'(2) where h = -3f - 3g. So h' = -3f' - 3g'. The equation computes -3g' - 3f'. This is the same value. Is there an error? Let's look closer. The equation is `-3*5 - 3*(-3)`. 5 is g'(2). -3 is f'(2). So it is `-3*g'(2) - 3*f'(2)`. This is equal to `h'(2)`. The result is -6. The stated answer is -6. The solution is correct. Why did I think it was wrong? I misread the mapping. Let's re-verify. h'(2) = -3f'(2) - 3g'(2) = -3(-3) - 3(5) = 9 - 15 = -6. The equation line is `-3*5 - 3*(-3) = -15 + 9 = -6`. It uses the values correctly, just in a different order (g term first, then f term). This is mathematically valid. The sentence is generic but true. Verdict: pass.qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution incorrectly substitutes f'(2) = -3 into the term for g'(x). The derivative is h'(x) = -3f'(x) - 3g'(x), so h'(2) = -3(-3) - 3(5) = 9 - 15 = -6. Wait, the calculation -3*5 - 3*(-3) implies the first term used g'(2)=5 and the second used f'(2)=-3, but the formula is -3f' - 3g'. The solution swapped the values: it calculated -3*g'(2) - 3*f'(2) instead of -3*f'(2) - 3*g'(2). Although the final numerical answer happens to be correct due to the specific values (-3*-3 - 3*5 = 9-15=-6 vs -3*5 -3*-3 = -15+9=-6), the substitution step shown is logically inconsistent with the stated formula h(x) = -3f(x) - 3g(x). Specifically, the equation line shows -3*5 (using g') for the first term and -3*-3 (using f') for the second term, effectively computing -3g'(2) - 3f'(2). This is a mismatch between the function definition and the substitution.gpt-oss:20b: pass 2026-10-07
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 2026-10-07 with SymPy 1.14.0.