The derivative from the limit definition
Problem 2.1532 · medium
Use the limit definition of the derivative to find \( \displaystyle f'(x) \) for \( \displaystyle f(x) = 2 x - 5 \), then find \( \displaystyle f'(2) \).
- By definition f′(x) = lim_{h→0} [f(x + h) − f(x)]/h.Reviewed
- \[ 2 h + 2 x - 5 \]Write out f(x + h).✓ Proved
- \[ 2 \]Combine and cancel the factor h.✓ Proved
- \[ \lim_{h \to 0^+} 2 = 2 \]Now h → 0 by direct substitution.✓ Proved
- \[ \left. 2 \right|_{\substack{ x=2 }} = 2 \]At x = 2.✓ Proved
Answer \( f'(x) = 2,\ f'(2) = 2 \)
Lines: 4 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 |
| 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: pass — The solution correctly applies the limit definition of the derivative to a linear function and evaluates the result at x=2. The steps are logically sound and the algebraic simplifications are correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the limit definition of the derivative to a linear function and evaluates the result at x=2. The steps are logically sound and the algebraic simplifications are correct.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to explicitly form the difference quotient [f(x+h) - f(x)]/h. It jumps from writing f(x+h) to the result 2, skipping the subtraction of f(x) and division by h, which are the core steps of the limit definition.gpt-oss:20b: fail (error) 2026-10-04 — Step 2 incorrectly writes f(x+h) as 2*h+2*x-5 but then fails to subtract f(x). The correct numerator is 2h, not the expression shown. The subsequent cancellation and limit are based on a faulty expression.
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.