The derivative from the limit definition
Problem 2.1786 · medium
Use the limit definition of the derivative to find \( \displaystyle f'(x) \) for \( \displaystyle f(x) = - \frac{3}{x} \).
- By definition f′(x) = lim_{h→0} [f(x + h) − f(x)]/h.Reviewed
- \[ - \frac{3}{h + x} \]Write out f(x + h).✓ Proved
- \[ \frac{- \frac{3}{h + x} + \frac{3}{x}}{h} = \frac{3}{x \left(h + x\right)} \]Combine and cancel the factor h.✓ Proved
- \[ \lim_{h \to 0^+}\left(\frac{3}{x \left(h + x\right)}\right) = \frac{3}{x^{2}} \]Now h → 0 by direct substitution.✓ Proved
Answer \( f'(x) = \frac{3}{x^{2}} \)
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, performs the algebraic simplification accurately, and evaluates the limit correctly.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly applies the limit definition of the derivative, performs the algebraic simplification accurately, and evaluates the limit correctly.gpt-oss:20b: fail (error) 2026-10-07 — The solution omits the subtraction of f(x) in step 2, so the difference quotient is not correctly formed. Consequently, the algebraic manipulation in step 3 is invalid, leading to an incorrect derivative.qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly applies the limit definition of the derivative, performs the algebraic simplification accurately, and evaluates the limit correctly.
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 2026-10-07 with SymPy 1.14.0.