The derivative from the limit definition
Problem 2.1525 · medium
Use the limit definition of the derivative to find \( \displaystyle f'(x) \) for \( \displaystyle f(x) = 4 - \frac{1}{x} \).
- By definition f′(x) = lim_{h→0} [f(x + h) − f(x)]/h.Reviewed
- \[ 4 - 1 \frac{1}{h + x} = 4 - \frac{1}{h + x} \]Write out f(x + h).✓ Proved
- \[ \frac{- \frac{1}{h + x} + \frac{1}{x}}{h} = \frac{1}{x \left(h + x\right)} \]Combine and cancel the factor h.✓ Proved
- \[ \lim_{h \to 0^+}\left(\frac{1}{x \left(h + x\right)}\right) = \frac{1}{x^{2}} \]Now h → 0 by direct substitution.✓ Proved
Answer \( f'(x) = \frac{1}{x^{2}} \)
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. The algebraic simplification and limit evaluation are accurate, and the final result matches the stated answer.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the limit definition of the derivative. The algebraic simplification and limit evaluation are accurate, and the final result matches the stated answer.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (error) 2026-10-04 — Line 2 is incorrect because it fails to subtract f(x) from f(x+h), resulting in an expression that does not represent the numerator of the difference quotient. Consequently, the algebraic simplification in Line 3 is based on a wrong premise.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.