Home›Calculus 1›Derivative rules from a table of values›Problem 2.2499

Derivative rules from a table of values

Problem 2.2499 · easy

Use the table to find \( \displaystyle h'(0) \) for \( \displaystyle h(x) = f(x) - 2 g(x) \). \[ \begin{array}{c|cccc} x & f(x) & f'(x) & g(x) & g'(x) \\ \hline 0 & -3 & -1 & 1 & 3\end{array} \]

Constant multiples and sums differentiate term by term.
\[ \left(-2\right) 3 + 1 \left(-1\right) = -7 \]
Read the values from the table and substitute.✓ Proved

Answer \( h'(0) = -7 \)

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. How we verify

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: pass
qwen3.6:27b-mlx: fail (error) — The solution substitutes f'(0) = -1 as 1*(-1), implying the coefficient of f'(0) is 1, but it omits the explicit differentiation step showing h'(x) = f'(x) - 2g'(x). While the arithmetic is correct, the sentence 'Constant multiples and sums differentiate term by term' is too vague to justify the specific coefficients used in the substitution without explicitly stating the derived formula for h'(x). A student might not see where the '1' comes from if the formula h'(x) = f'(x) - 2g'(x) is not explicitly written.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-26 — The solution substitutes f'(0) = -1 as 1*(-1), implying the coefficient of f'(0) is 1, but it omits the explicit differentiation step showing h'(x) = f'(x) - 2g'(x). While the arithmetic is correct, the sentence 'Constant multiples and sums differentiate term by term' is too vague to justify the specific coefficients used in the substitution without explicitly stating the derived formula for h'(x). A student might not see where the '1' comes from if the formula h'(x) = f'(x) - 2g'(x) is not explicitly written.
gpt-oss:20b: pass 2026-10-26
qwen3.6:27b-mlx: fail (error) 2026-10-25 — The solution substitutes f'(0) = -1 as 1*(-1), implying a coefficient of 1 for f'(x), but the derivative of h(x) is f'(x) - 2g'(x). The term for f'(x) should just be -1, not 1*(-1) if interpreted as multiplying by a coefficient from the original function (though 1 is the coefficient, the phrasing '1*(-1)' is confusingly juxtaposed with '-2*3'). More critically, the sentence 'Constant multiples and sums differentiate term by term' is vague but acceptable, however, the equation line shows '-2*3 + 1*(-1)'. The derivative is h'(x) = f'(x) - 2g'(x). At x=0, h'(0) = f'(0) - 2g'(0) = -1 - 2(3) = -1 - 6 = -7. The calculation -2*3 + 1*(-1) equals -6 - 1 = -7. The arithmetic is correct. Wait, let me re-read carefully. h(x) = f(x) - 2g(x). h'(x) = f'(x) - 2g'(x). h'(0) = f'(0) - 2g'(0). From table: f'(0) = -1, g'(0) = 3. So h'(0) = -1 - 2(3) = -7. The solution writes '-2*3 + 1*(-1)'. This corresponds to -2*g'(0) + 1*f'(0). This is mathematically correct. Is there an error? The sentence says 'Constant multiples and sums differentiate term by term'. This is a bit loose but standard. The equation is marked 'proved' by CAS. The substitution is correct. Why would it fail? Let's look closer. 'Read the values from the table and substitute.' is part of the equation line in the prompt description? No, the prompt says '2. [equation, proved] ... -- Read the values...'. The text after the equation is likely a comment. The equation itself is -2*3 + 1*(-1) = -7. This is correct. Is there a subtle error? h(x) = f(x) - 2g(x). h'(x) = f'(x) - 2g'(x). The solution calculates -2*3 + 1*(-1). This is -6 - 1 = -7. Correct. Maybe the 'error' is that the sentence doesn't explicitly state the derivative rule h'(x) = f'(x) - 2g'(x)? It just says 'differentiate term by term'. That's a bit weak but not an error. Let's look for a 'misleading' aspect. The coefficient of f(x) is 1. The solution writes 1*(-1). This is fine. What if the user considers the lack of explicit derivative formula an error? The prompt asks to review sentences. Sentence 1: 'Constant multiples and sums differentiate term by term.' This is a valid heuristic. Sentence 2 is an equation. The comment 'Read the values...' is not a sentence to be reviewed for logic, it's a label. Wait, the prompt says 'Each line... is either an EQUATION... or a SENTENCE'. Line 2 is an equation. The text '-- Read the values...' is likely part of the equation line's metadata or a comment. If it's treated as a sentence, it's fine. Is there a sign error? f'(0) = -1. g'(0) = 3. h'(0) = -1 - 6 = -7. The solution has -2*3 + 1*(-1) = -6 - 1 = -7. It seems correct. Why would I fail it? Maybe the order? No. Maybe the sentence is too vague? 'Constant multiples and sums differentiate term by term' is not a precise theorem statement. It's a description of linearity. It's acceptable. Let's look really closely. h(x) = f(x) - 2g(x). The derivative is f'(x) - 2g'(x). The solution computes -2*3 + 1*(-1). This is -2g'(0) + f'(0). This is correct. Is it possible the problem implies h(x) = f(x) - 2g(x) means the coefficient of g is -2? Yes. So -2 * g'(0) is correct. And 1 * f'(0) is correct. I see no error. Wait, look at the table. f'(0) is -1. g'(0) is 3. The solution uses 3 and -1. Correct. Is there a trap? 'Use the table to find h'(0)'. The solution does that. I will pass it. Actually, let me double check the 'proved' status. The CAS checked the algebra. -2*3 + 1*(-1) = -7 is true. The mapping from h'(0) to that expression is the logical step. The sentence supports it. It seems sound. However, often these 'adversarial' reviews catch missing steps. The step from h(x) to h'(x) is skipped in the sentences. Sentence 1 says 'differentiate term by term'. It doesn't explicitly write h'(x) = f'(x) - 2g'(x). Is that an error? It's a 'style' or 'misleading' issue if a student doesn't know the rule. But 'differentiate term by term' is the standard name for the linearity rule. I'll give it a pass. Wait, looking at similar examples, sometimes 'differentiate term by term' is considered insufficient if the specific rule (product, chain, etc.) is needed. Here it's just linearity. It's fine. Let me look for a sign error again. f(x) - 2g(x). Derivative: f'(x) - 2g'(x). Values: -1 - 2(3) = -7. Solution: -2(3) + 1(-1) = -7. Correct. I will pass.
gpt-oss:20b: pass 2026-10-25
qwen3.6:27b-mlx: fail (error) 2026-10-25 — The solution fails to explicitly state the differentiation rule h'(x) = f'(x) - 2g'(x) before substituting values. While the arithmetic is correct, omitting the symbolic derivative step makes the substitution unjustified and skips the core calculus concept required by the problem.
gpt-oss:20b: pass 2026-10-25

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 2026-10-26 with SymPy 1.14.0.