The derivative from the limit definition
Problem 2.1794 · medium
Use the limit definition of the derivative to find \( \displaystyle f'(x) \) for \( \displaystyle f(x) = - x^{3} - 4 x \), then find \( \displaystyle f'(1) \).
- By definition f′(x) = lim_{h→0} [f(x + h) − f(x)]/h.
- \[ - 4 h - 4 x - \left(h + x\right)^{3} = - h^{3} - 3 h^{2} x - 3 h x^{2} - 4 h - x^{3} - 4 x \]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
- \[ \left. - 3 x^{2} - 4 \right|_{\substack{ x=1 }} = -7 \]At x = 1.✓ Proved
Answer \( f'(x) = - 3 x^{2} - 4,\ f'(1) = -7 \)
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: passqwen3.6:27b-mlx: fail (error) — Step 2 is labeled 'Write out f(x + h)' but the equation shown is actually the numerator f(x+h) - f(x). Step 3 is labeled 'Combine and cancel the factor h' but the equation shown is the result of that division (the difference quotient), not the combination step itself. The labels do not match the mathematical content of the equations.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — Step 2 is labeled 'Write out f(x + h)' but the equation shown is actually the numerator f(x+h) - f(x). Step 3 is labeled 'Combine and cancel the factor h' but the equation shown is the result of that division (the difference quotient), not the combination step itself. The labels do not match the mathematical content of the equations.gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The setup in step 2 is incorrect. The expression for f(x+h) is missing the cubic term for x (it should be -(x+h)^3 - 4(x+h)), and the subtraction of f(x) is handled incorrectly, leading to an algebraically invalid intermediate step. Specifically, f(x+h) - f(x) simplifies to -3x^2h - 3xh^2 - h^3 - 4h, but the solution's left-hand side in step 2 does not represent f(x+h) - f(x) correctly.
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 2026-10-07 with SymPy 1.14.0.