The derivative from the limit definition
Problem 2.2094 · medium
Use the limit definition of the derivative to find \( \displaystyle f'(x) \) for \( \displaystyle f(x) = - 3 x^{2} + 4 x + 5 \).
- By definition f′(x) = lim_{h→0} [f(x + h) − f(x)]/h.Reviewed
- \[ 4 h + 4 x - 3 \left(h + x\right)^{2} + 5 = - 3 h^{2} - 6 h x + 4 h - 3 x^{2} + 4 x + 5 \]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
- \[ \lim_{h \to 0^+}\left(- 3 h - 6 x + 4\right) = 4 - 6 x \]Now h → 0 by direct substitution.✓ Proved
Answer \( f'(x) = 4 - 6 x \)
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: passqwen3.6:27b-mlx: pass — The solution correctly applies the limit definition of the derivative. The algebraic steps are verified, and the final limit evaluation is correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly applies the limit definition of the derivative. The algebraic steps are verified, and the final limit evaluation is correct.gpt-oss:20b: pass 2026-10-11gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution fails to explicitly compute f(x+h) and f(x) separately before subtracting, making the transition from step 1 to step 2 opaque and potentially confusing. Step 2 presents an equality that mixes the expression for f(x+h) with terms from f(x) in a way that does not clearly show the numerator of the difference quotient, violating the standard structure of the limit definition proof.
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 2026-10-11 with SymPy 1.14.0.