The derivative from the limit definition
Problem 2.1945 · medium
Use the limit definition of the derivative to find \( \displaystyle f'(x) \) for \( \displaystyle f(x) = 4 - 2 x \), 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 fails to explicitly form the difference quotient [f(x+h) - f(x)]/h as required by the limit definition. It jumps from writing f(x+h) to the simplified result, skipping the crucial algebraic step of subtracting f(x) and dividing by h.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution fails to explicitly form the difference quotient [f(x+h) - f(x)]/h as required by the limit definition. It jumps from writing f(x+h) to the simplified result, skipping the crucial algebraic step of subtracting f(x) and dividing by h.qwen3.6:27b-mlx: fail (error) 2026-10-09 — The solution skips the crucial step of forming the difference quotient [f(x+h) - f(x)]/h. It jumps from writing f(x+h) to the simplified result, failing to demonstrate the subtraction and division by h required by 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.