The derivative from the limit definition
Problem 2.2092 · medium
Use the limit definition of the derivative to find \( \displaystyle f'(x) \) for \( \displaystyle f(x) = x - 1 \), then find \( \displaystyle f'(1) \).
- By definition f′(x) = lim_{h→0} [f(x + h) − f(x)]/h.
- \[ h + x - 1 \]Write out f(x + h).✓ Proved
- \[ 1 \]Combine and cancel the factor h.✓ Proved
- \[ \lim_{h \to 0^+} 1 = 1 \]Now h → 0 by direct substitution.✓ Proved
- \[ \left. 1 \right|_{\substack{ x=1 }} = 1 \]At x = 1.✓ Proved
Answer \( f'(x) = 1,\ f'(1) = 1 \)
Lines: 4 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 |
| 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: passqwen3.6:27b-mlx: fail (error) — The solution fails to actually apply the limit definition. It skips the crucial step of forming the difference quotient [f(x+h) - f(x)]/h and instead just asserts f(x+h) equals itself, never performing the subtraction or division by h required by the definition.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution fails to actually apply the limit definition. It skips the crucial step of forming the difference quotient [f(x+h) - f(x)]/h and instead just asserts f(x+h) equals itself, never performing the subtraction or division by h required by the definition.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 apply the limit definition correctly. It computes f(x+h) but never subtracts f(x) or divides by h, effectively skipping the core mechanism of the difference quotient.
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.