Home›Calculus 1›Derivative rules from a table of values›Problem 2.2099

Derivative rules from a table of values

Problem 2.2099 · easy

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

Constant multiples and sums differentiate term by term.
\[ 4 \left(-3\right) - 2 \left(-2\right) = -8 \]
Read the values from the table and substitute.✓ Proved

Answer \( h'(2) = -8 \)

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: pass
qwen3.6:27b-mlx: fail (error) — The solution skips the crucial step of stating the derivative formula h'(x) = -2f'(x) + 4g'(x) before substitution. It jumps directly from the rule to the arithmetic, failing to explicitly show how the table values map to the derivative expression.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution skips the crucial step of stating the derivative formula h'(x) = -2f'(x) + 4g'(x) before substitution. It jumps directly from the rule to the arithmetic, failing to explicitly show how the table values map to the derivative expression.
gpt-oss:20b: pass 2026-10-11
qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly applies the linearity of the derivative operator and substitutes the values from the table accurately.
gpt-oss:20b: pass 2026-10-11

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 2026-10-11 with SymPy 1.14.0.