The derivative from the limit definition
Problem 2.1950 · medium
Use the limit definition of the derivative to find \( \displaystyle f'(x) \) for \( \displaystyle f(x) = 2 x - 4 \), then find \( \displaystyle f'(-1) \).
- By definition f′(x) = lim_{h→0} [f(x + h) − f(x)]/h.
- \[ 2 h + 2 x - 4 \]Write out f(x + h).✓ Proved
- \[ 2 \]Combine and cancel the factor h.✓ Proved
- \[ \lim_{h \to 0^+} 2 = 2 \]Now h → 0 by direct substitution.✓ Proved
- \[ \left. 2 \right|_{\substack{ x=-1 }} = 2 \]At x = -1.✓ Proved
Answer \( f'(x) = 2,\ f'(-1) = 2 \)
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) — The solution skips the critical step of forming the difference quotient [f(x+h) - f(x)]/h. Line 2 merely states f(x+h), and Line 3 jumps to the result 2 without showing the subtraction of f(x) or the division by h, which is the core of the limit definition method.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution skips the critical step of forming the difference quotient [f(x+h) - f(x)]/h. Line 2 merely states f(x+h), and Line 3 jumps to the result 2 without showing the subtraction of f(x) or the division by h, which is the core of the limit definition method.qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution fails to actually compute the difference quotient [f(x+h) - f(x)]/h. It merely writes out f(x+h) and then jumps to the result 2 without showing the subtraction of f(x) or the division by h, which are the core steps of the limit definition.gpt-oss:20b: pass 2026-10-09
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 2026-10-09 with SymPy 1.14.0.