The derivative from the limit definition
Problem 2.1530 · medium
Use the limit definition of the derivative to find \( \displaystyle f'(x) \) for \( \displaystyle f(x) = x^{3} + 4 x \).
- By definition f′(x) = lim_{h→0} [f(x + h) − f(x)]/h.Reviewed
- \[ 4 h + 4 x + \left(h + x\right)^{3} = h^{3} + 3 h^{2} x + 3 h x^{2} + 4 h + x^{3} + 4 x \]Write out f(x + h).✓ Proved
- \[ \frac{4 h - x^{3} + \left(h + x\right)^{3}}{h} = h^{2} + 3 h x + 3 x^{2} + 4 \]Combine and cancel the factor h.✓ Proved
- \[ \lim_{h \to 0^+}\left(h^{2} + 3 h x + 3 x^{2} + 4\right) = 3 x^{2} + 4 \]Now h → 0 by direct substitution.✓ Proved
Answer \( f'(x) = 3 x^{2} + 4 \)
✓ 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.
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, with valid algebraic simplifications and correct limit evaluation.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the limit definition of the derivative, with valid algebraic simplifications and correct limit evaluation.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the limit definition of the derivative, performs the algebraic simplification accurately, and evaluates the limit correctly.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.