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The derivative from the limit definition

Problem 2.1527 · medium

Use the limit definition of the derivative to find \( \displaystyle f'(x) \) for \( \displaystyle f(x) = 3 x^{2} - 4 x - 4 \), then find \( \displaystyle f'(2) \).

By definition f′(x) = lim_{h→0} [f(x + h) − f(x)]/h.
Reviewed
\[ - 4 h - 4 x + 3 \left(h + x\right)^{2} - 4 = 3 h^{2} + 6 h x - 4 h + 3 x^{2} - 4 x - 4 \]
Write out f(x + h).✓ Proved
\[ \frac{- 4 h - 3 x^{2} + 3 \left(h + x\right)^{2}}{h} = 3 h + 6 x - 4 \]
Combine and cancel the factor h.✓ Proved
\[ \lim_{h \to 0^+}\left(3 h + 6 x - 4\right) = 6 x - 4 \]
Now h → 0 by direct substitution.✓ Proved
\[ \left. 6 x - 4 \right|_{\substack{ x=2 }} = 8 \]
At x = 2.✓ Proved

Answer \( f'(x) = 6 x - 4,\ f'(2) = 8 \)

Lines: 4 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
5	✓ 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, performs the algebraic simplification accurately, and evaluates the result correctly.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the limit definition of the derivative, performs the algebraic simplification accurately, and evaluates the result correctly.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The algebraic expansion in step 2 is incorrect; it fails to distribute the negative sign to f(x) when computing f(x+h) - f(x), resulting in an incorrect numerator. Consequently, the simplification in step 3 is based on this error, although it coincidentally yields the correct derivative formula.
gpt-oss:20b: pass 2026-10-04

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 2026-10-04 with SymPy 1.14.0.