The derivative from the limit definition
Problem 2.2490 · medium Video
Use the limit definition of the derivative to find \( \displaystyle f'(x) \) for \( \displaystyle f(x) = 4 x + 6 \).
- By definition f′(x) = lim_{h→0} [f(x + h) − f(x)]/h.Reviewed
- \[ \left. 4 x + 6 \right|_{\substack{ x=h + x }} = 4 h + 4 x + 6 \]Write out f(x + h).✓ Proved
- \[ \frac{4 h + 4 x - \left(4 x + 6\right) + 6}{h} = 4 \]Combine and cancel the factor h.✓ Proved
- \[ \lim_{h \to 0^+} 4 = 4 \]Now h → 0 by direct substitution.✓ Proved
Answer \( f'(x) = 4 \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify
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 is accurate, and the final limit evaluation is correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-26 — The solution correctly applies the limit definition of the derivative. The algebraic simplification is accurate, and the final limit evaluation is correct.gpt-oss:20b: pass 2026-10-26qwen3.6:27b-mlx: pass 2026-10-25 — The solution correctly applies the limit definition of the derivative. The algebraic simplification and limit evaluation are correct.gpt-oss:20b: pass 2026-10-25qwen3.6:27b-mlx: pass 2026-10-25 — The solution correctly applies the limit definition of the derivative. The algebraic simplification and limit evaluation are correct.gpt-oss:20b: pass 2026-10-25
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 2026-10-26 with SymPy 1.14.0.