Home›Calculus 1›Derivative rules from a table of values›Problem 2.1648

Derivative rules from a table of values

Problem 2.1648 · easy

Use the table to find \( \displaystyle h'(0) \) for \( \displaystyle h(x) = f(x) g(x) \). \[ \begin{array}{c|cccc} x & f(x) & f'(x) & g(x) & g'(x) \\ \hline 0 & 6 & -2 & 3 & -4\end{array} \]

Product rule: h′ = f′g + fg′.
Reviewed
\[ 6 \left(-4\right) - 2 \cdot 3 = -30 \]
Read the values from the table and substitute.✓ Proved

Answer \( h'(0) = -30 \)

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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the product rule and substitutes the values from the table accurately.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies the product rule and substitutes the values from the table accurately.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The substitution in step 2 is incorrect. The product rule is h'(x) = f'(x)g(x) + f(x)g'(x). At x=0, this is (-2)(3) + (6)(-4) = -6 - 24 = -30. The solution writes 6*(-4) - 2*3, which corresponds to f(x)g'(x) - f'(x)g(x) (the quotient rule numerator structure or just a sign error on the first term), but coincidentally yields the correct number. However, the expression written does not match the stated rule 'f'g + fg''. Specifically, it calculates fg' - f'g instead of f'g + fg'. While the final number is right, the algebraic representation of the rule application is wrong.
gpt-oss:20b: pass 2026-10-05

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 2026-10-05 with SymPy 1.14.0.