Home›Calculus 1›The derivative from the limit definition›Problem 2.1791

The derivative from the limit definition

Problem 2.1791 · medium

Use the limit definition of the derivative to find \( \displaystyle f'(x) \) for \( \displaystyle f(x) = 2 x - 6 \).

By definition f′(x) = lim_{h→0} [f(x + h) − f(x)]/h.
Reviewed
\[ 2 h + 2 x - 6 \]
Write out f(x + h).✓ Proved
\[ 2 \]
Combine and cancel the factor h.✓ Proved
\[ \lim_{h \to 0^+} 2 = 2 \]
Now h → 0 by direct substitution.✓ Proved

Answer \( f'(x) = 2 \)

Lines: 3 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
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	✓ 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	matches the derivative by the power, quotient and chain rules

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the limit definition of the derivative. The algebraic simplification and limit evaluation are correct, leading to the right result.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly applies the limit definition of the derivative. The algebraic simplification and limit evaluation are correct, leading to the right result.
gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to actually compute the difference quotient [f(x+h) - f(x)]/h. It merely restates f(x+h) and then jumps to the result without showing the subtraction of f(x) or the division by h, which are the core steps of the limit definition.

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_definition, checked 2026-10-07 with SymPy 1.14.0.