The derivative from the limit definition
Problem 2.1793 · medium
Use the limit definition of the derivative to find \( \displaystyle f'(x) \) for \( \displaystyle f(x) = - 4 x - 4 \), then find \( \displaystyle f'(-1) \).
- By definition f′(x) = lim_{h→0} [f(x + h) − f(x)]/h.
- \[ - 4 h - 4 x - 4 \]Write out f(x + h).✓ Proved
- \[ -4 \]Combine and cancel the factor h.✓ Proved
- \[ \lim_{h \to 0^+} -4 = -4 \]Now h → 0 by direct substitution.✓ Proved
- \[ \left. -4 \right|_{\substack{ x=-1 }} = -4 \]At x = -1.✓ Proved
Answer \( f'(x) = -4,\ f'(-1) = -4 \)
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) — Step 2 incorrectly writes f(x+h); it should be -4(x+h)-4 = -4x-4h-4, not the same expression on both sides. Consequently step 3’s cancellation of h is invalid.qwen3.6:27b-mlx: fail (error) — The solution fails to actually compute f(x+h) in the difference quotient, jumping straight to the result without showing the subtraction f(x+h) - f(x). Step 2 is algebraically nonsensical as a representation of f(x+h) and does not lead to the difference quotient.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-07 — Step 2 incorrectly writes f(x+h); it should be -4(x+h)-4 = -4x-4h-4, not the same expression on both sides. Consequently step 3’s cancellation of h is invalid.qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to actually compute f(x+h) in the difference quotient, jumping straight to the result without showing the subtraction f(x+h) - f(x). Step 2 is algebraically nonsensical as a representation of f(x+h) and does not lead to the difference quotient.gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to actually compute f(x+h) and f(x) in the difference quotient. Line 2 presents an algebraic identity that does not represent the numerator f(x+h) - f(x), skipping the essential steps of substitution and subtraction required by the limit definition.
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.