The derivative from the limit definition
Problem 2.1639 · medium
Use the limit definition of the derivative to find \( \displaystyle f'(x) \) for \( \displaystyle f(x) = -2 - \frac{4}{x} \), then find \( \displaystyle f'(2) \).
- By definition f′(x) = lim_{h→0} [f(x + h) − f(x)]/h.Reviewed
- \[ -2 - \frac{4}{h + x} \]Write out f(x + h).✓ Proved
- \[ \frac{- \frac{4}{h + x} + \frac{4}{x}}{h} = \frac{4}{x \left(h + x\right)} \]Combine and cancel the factor h.✓ Proved
- \[ \lim_{h \to 0^+}\left(\frac{4}{x \left(h + x\right)}\right) = \frac{4}{x^{2}} \]Now h → 0 by direct substitution.✓ Proved
- \[ \left. \frac{4}{x^{2}} \right|_{\substack{ x=2 }} = 1 \]At x = 2.✓ Proved
Answer \( f'(x) = \frac{4}{x^{2}},\ f'(2) = 1 \)
✓ Nihil obstat Lines: 4 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 |
| 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, performs the algebraic simplification accurately, and evaluates the result at x=2. The steps are logically sound and the final answer is correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies the limit definition of the derivative, performs the algebraic simplification accurately, and evaluates the result at x=2. The steps are logically sound and the final answer is correct.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies the limit definition of the derivative, performs the algebraic simplification accurately, and evaluates the result at x=2.gpt-oss:20b: pass 2026-10-05
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 2026-10-05 with SymPy 1.14.0.