The derivative from the limit definition
Problem 2.1787 · medium
Use the limit definition of the derivative to find \( \displaystyle f'(x) \) for \( \displaystyle f(x) = - 3 x - 2 \), then find \( \displaystyle f'(1) \).
- By definition f′(x) = lim_{h→0} [f(x + h) − f(x)]/h.
- \[ - 3 h - 3 x - 2 \]Write out f(x + h).✓ Proved
- \[ -3 \]Combine and cancel the factor h.✓ Proved
- \[ \lim_{h \to 0^+} -3 = -3 \]Now h → 0 by direct substitution.✓ Proved
- \[ \left. -3 \right|_{\substack{ x=1 }} = -3 \]At x = 1.✓ Proved
Answer \( f'(x) = -3,\ f'(1) = -3 \)
Lines: 4 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see 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 (misleading) — Step 2 incorrectly states f(x+h) without subtracting f(x); the cancellation step is not justified until the subtraction is shown. The student could be misled into thinking the derivative calculation is trivial from the expression alone.qwen3.6:27b-mlx: fail (error) — Line 2 is labeled 'Write out f(x + h)' but the equation shown is an identity (-3h - 3x - 2 = -3h - 3x - 2) that does not represent f(x+h) alone, nor does it show the subtraction of f(x) required by the definition in Line 1. The setup for the limit definition is missing the numerator construction f(x+h) - f(x).
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-07 — Step 2 incorrectly states f(x+h) without subtracting f(x); the cancellation step is not justified until the subtraction is shown. The student could be misled into thinking the derivative calculation is trivial from the expression alone.qwen3.6:27b-mlx: fail (error) 2026-10-07 — Line 2 is labeled 'Write out f(x + h)' but the equation shown is an identity (-3h - 3x - 2 = -3h - 3x - 2) that does not represent f(x+h) alone, nor does it show the subtraction of f(x) required by the definition in Line 1. The setup for the limit definition is missing the numerator construction f(x+h) - f(x).gpt-oss:20b: fail (misleading) 2026-10-07 — The solution skips the crucial step of computing f(x+h)-f(x); it merely states an identity and then claims to cancel h, which is not justified and could mislead a student about how the limit is evaluated.qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to actually compute f(x+h) and f(x+h)-f(x) as required by the limit definition; it skips the algebraic setup entirely and jumps to the result. Additionally, the unchecked equations are malformed and do not represent the steps of the derivation.
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.