Home›Calculus 1›Derivative rules from a table of values›Problem 2.1641

Derivative rules from a table of values

Problem 2.1641 · easy

Use the table to find \( \displaystyle h'(3) \) for \( \displaystyle h(x) = x f(x) + \left[g(x)\right]^{2} \). \[ \begin{array}{c|cccc} x & f(x) & f'(x) & g(x) & g'(x) \\ \hline 3 & -5 & 4 & 4 & -1\end{array} \]

Product rule on x·f(x), chain rule on g(x)².
Reviewed
\[ 2 \cdot 4 \left(-1\right) - 5 + 3 \cdot 4 = -1 \]
Read the values from the table and substitute.✓ Proved

Answer \( h'(3) = -1 \)

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 and chain rules, substitutes the values from the table accurately, and arrives at the correct result.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies the product and chain rules, substitutes the values from the table accurately, and arrives at the correct result.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to explicitly state the derivative formula h'(x) = f(x) + x f'(x) + 2g(x)g'(x) before substitution. While the arithmetic in line 2 is correct, omitting the symbolic differentiation step makes it impossible to verify if the student correctly applied the product and chain rules, rather than just guessing the numbers.
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.