The derivative from the limit definition
Problem 2.2090 · medium
Use the limit definition of the derivative to find \( \displaystyle f'(x) \) for \( \displaystyle f(x) = - 3 x^{2} - 2 x - 3 \).
- By definition f′(x) = lim_{h→0} [f(x + h) − f(x)]/h.Reviewed
- \[ - 2 h - 2 x - 3 \left(h + x\right)^{2} - 3 = - 3 h^{2} - 6 h x - 2 h - 3 x^{2} - 2 x - 3 \]Write out f(x + h).✓ Proved
- \[ \frac{- 2 h + 3 x^{2} - 3 \left(h + x\right)^{2}}{h} = - 3 h - 6 x - 2 \]Combine and cancel the factor h.✓ Proved
- \[ \lim_{h \to 0^+}\left(- 3 h - 6 x - 2\right) = - 6 x - 2 \]Now h → 0 by direct substitution.✓ Proved
Answer \( f'(x) = - 6 x - 2 \)
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: passqwen3.6:27b-mlx: pass — The solution correctly applies the limit definition of the derivative. Although the intermediate algebraic steps are presented in a condensed manner, the equations are marked as proved and are mathematically correct, leading to the right final answer.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly applies the limit definition of the derivative. Although the intermediate algebraic steps are presented in a condensed manner, the equations are marked as proved and are mathematically correct, leading to the right final answer.gpt-oss:20b: pass 2026-10-11gpt-oss:20b: fail (error) 2026-10-11 — The subtraction f(x+h)-f(x) is performed incorrectly; the terms -3*x**2, -2*x, and -3 should not appear in the difference. Consequently the simplification and limit step are based on a wrong expression.qwen3.6:27b-mlx: fail (error) 2026-10-11 — Step 2 incorrectly computes f(x+h) as -3(h+x)^2 - 2h - 2x - 3, but the constant term -3 is missing from the expression for f(x+h) (it should be -3(h+x)^2 - 2(h+x) - 3). Although the error cancels out in the numerator f(x+h)-f(x) because the constant -3 is present in both f(x+h) and f(x), the line explicitly claims to 'Write out f(x+h)' and provides an incorrect expression for f(x+h) itself. This is a factual error in the intermediate step.
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.