The derivative from the limit definition
Problem 2.1528 · medium
Use the limit definition of the derivative to find \( \displaystyle f'(x) \) for \( \displaystyle f(x) = 2 x^{3} - 5 x \), then find \( \displaystyle f'(1) \).
- By definition f′(x) = lim_{h→0} [f(x + h) − f(x)]/h.
- \[ - 5 h - 5 x + 2 \left(h + x\right)^{3} = 2 h^{3} + 6 h^{2} x + 6 h x^{2} - 5 h + 2 x^{3} - 5 x \]Write out f(x + h).✓ Proved
- \[ \frac{- 5 h - 2 x^{3} + 2 \left(h + x\right)^{3}}{h} = 2 h^{2} + 6 h x + 6 x^{2} - 5 \]Combine and cancel the factor h.✓ Proved
- \[ \lim_{h \to 0^+}\left(2 h^{2} + 6 h x + 6 x^{2} - 5\right) = 6 x^{2} - 5 \]Now h → 0 by direct substitution.✓ Proved
- \[ \left. 6 x^{2} - 5 \right|_{\substack{ x=1 }} = 1 \]At x = 1.✓ Proved
Answer \( f'(x) = 6 x^{2} - 5,\ f'(1) = 1 \)
Lines: 4 proved, 1 not checked. 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 | Not checked | — | 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: fail (error) — The expression for f(x+h)−f(x) is incorrect: it omits the +5x term from f(x) and therefore the subsequent algebra is based on a wrong numerator.qwen3.6:27b-mlx: pass — The solution correctly applies the limit definition of the derivative, performs the algebraic simplification accurately, and evaluates the result at x=1.
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 x=1.gpt-oss:20b: fail (error) 2026-10-04 — The expression for f(x+h)−f(x) is incorrect: it omits the +5x term from f(x) and therefore the subsequent algebra is based on a wrong numerator.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 x=1 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.