The derivative from the limit definition
Problem 2.1529 · medium
Use the limit definition of the derivative to find \( \displaystyle f'(x) \) for \( \displaystyle f(x) = \frac{1}{x^{2}} \), then find \( \displaystyle f'(-1) \).
- By definition f′(x) = lim_{h→0} [f(x + h) − f(x)]/h.Reviewed
- \[ \frac{1}{\left(h + x\right)^{2}} \]Write out f(x + h).✓ Proved
- \[ \frac{\frac{1}{\left(h + x\right)^{2}} - \frac{1}{x^{2}}}{h} = \frac{- h - 2 x}{x^{2} \left(h + x\right)^{2}} \]Combine and cancel the factor h.✓ Proved
- \[ \lim_{h \to 0^+}\left(\frac{- h - 2 x}{x^{2} \left(h + x\right)^{2}}\right) = - \frac{2}{x^{3}} \]Now h → 0 by direct substitution.✓ Proved
- \[ \left. - \frac{2}{x^{3}} \right|_{\substack{ x=-1 }} = 2 \]At x = -1.✓ Proved
Answer \( f'(x) = - \frac{2}{x^{3}},\ f'(-1) = 2 \)
✓ 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 the specified point.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the limit definition of the derivative, performs the algebraic simplification accurately, and evaluates the result at the specified point.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 result 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.