The derivative from the limit definition
Problem 2.1951 · medium
Use the limit definition of the derivative to find \( \displaystyle f'(x) \) for \( \displaystyle f(x) = - x^{3} - 3 x \).
- By definition f′(x) = lim_{h→0} [f(x + h) − f(x)]/h.
- \[ - 3 h - 3 x - \left(h + x\right)^{3} = - h^{3} - 3 h^{2} x - 3 h x^{2} - 3 h - x^{3} - 3 x \]Write out f(x + h).✓ Proved
- \[ \frac{- 3 h + x^{3} - \left(h + x\right)^{3}}{h} = - h^{2} - 3 h x - 3 x^{2} - 3 \]Combine and cancel the factor h.✓ Proved
- \[ \lim_{h \to 0^+}\left(- h^{2} - 3 h x - 3 x^{2} - 3\right) = - 3 x^{2} - 3 \]Now h → 0 by direct substitution.✓ Proved
Answer \( f'(x) = - 3 x^{2} - 3 \)
Lines: 3 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 |
| 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: fail (error) — Line 2 incorrectly expands f(x+h) as -3h - 3x - (h+x)^3, omitting the negative sign for the cubic term in the original function f(x) = -x^3 - 3x. Consequently, Line 3 uses an incorrect numerator (-3h + x^3 - ...) instead of the correct difference f(x+h) - f(x). Although the final answer is correct, the intermediate algebraic steps are wrong.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: fail (error) 2026-10-09 — Line 2 incorrectly expands f(x+h) as -3h - 3x - (h+x)^3, omitting the negative sign for the cubic term in the original function f(x) = -x^3 - 3x. Consequently, Line 3 uses an incorrect numerator (-3h + x^3 - ...) instead of the correct difference f(x+h) - f(x). Although the final answer is correct, the intermediate algebraic steps are wrong.qwen3.6:27b-mlx: inconclusive 2026-10-09 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The algebraic expansion in step 2 is incorrect; it expands -(x+h)^3 as -x^3 - 3x^2h - 3xh^2 - h^3 but incorrectly includes a term -3x (likelygpt-oss:20b: inconclusive 2026-10-09 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The expansion of \(f(x+h)\) is incorrect: it omits the \(-3(x+h)\) term, leading to a wrong algebraic simplification and an incorrect derivative."}
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 2026-10-09 with SymPy 1.14.0.