The derivative from the limit definition
Problem 2.1790 · medium
Use the limit definition of the derivative to find \( \displaystyle f'(x) \) for \( \displaystyle f(x) = 4 x + 6 \), then find \( \displaystyle f'(2) \).
- By definition f′(x) = lim_{h→0} [f(x + h) − f(x)]/h.Reviewed
- \[ 4 h + 4 x + 6 \]Write out f(x + h).✓ Proved
- \[ 4 \]Combine and cancel the factor h.✓ Proved
- \[ \lim_{h \to 0^+} 4 = 4 \]Now h → 0 by direct substitution.✓ Proved
- \[ \left. 4 \right|_{\substack{ x=2 }} = 4 \]At x = 2.✓ Proved
Answer \( f'(x) = 4,\ f'(2) = 4 \)
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, resulting in the correct derivative and evaluation.
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 to a linear function, resulting in the correct derivative and evaluation.gpt-oss:20b: fail (error) 2026-10-07 — Step 2 incorrectly writes f(x+h) without subtracting f(x); the difference f(x+h)-f(x) should be 4h, not the expression shown. This mis‑states the algebra needed for the derivative.qwen3.6:27b-mlx: fail (error) 2026-10-07 — Step 2 incorrectly identifies f(x+h) as 4h + 4x + 6, missing the multiplication by x (it should be 4(x+h) + 6 = 4x + 4h + 6). Although the final answer is correct, the intermediate algebraic setup is wrong.
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.