Derivative of \( \displaystyle \ln{\left(x - 3 \right)} \)
Problem 2.717 · easy Mental math Video
Differentiate \( \displaystyle f(x) = \ln{\left(x - 3 \right)} \).
- \[ \frac{d}{d x} \ln{\left(x - 3 \right)} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{1}{x - 3} \]derivativeApply the derivative rule for the natural logarithm.✓ Proved
Answer \( \frac{1}{x - 3} \)
Mind the domain. The answer is also defined on (-oo, 3), where f(x) is not. Substituting there gives a number that is not a slope of f.
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 3 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where x - 3 = 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: fail (style) — [domain objection, downgraded to style] Step 1 does not change the expression and is labeled with an undefined rule. It should either be omitted or replaced with a proper derivative step that applies the logarithm rule.qwen3.6:27b-mlx: fail (error) — Step 2 applies the derivative of the natural logarithm but fails to apply the chain rule for the inner function (x - 3). While the algebraic result is coincidentally correct because the derivative of the inner function is 1, the step claims to apply only the 'derivative' rule (unfolding d/dx on a known form) and ignores the composition, violating the single-rule-per-step constraint and mislabeling the operation.
Senior review claude-sonnet-5-5, 2026-10-12: pass — d/dx log(x-3) = 1/(x-3) is correct and the single step is labelled with a legitimate rule. SymPy proved every line and the final answer.
gpt-oss:20b: dismiss — Line 1 is the problem statement, which is not a step and needs no rule label, and omitting the derivative step is not required; the objection is also domain-based and not a real defect.qwen3.6:27b-mlx: dismiss — Evaluating the derivative of a simple inner function (x - 3 has derivative 1) is part of the step, and 'derivative' is a legitimate label for d/dx log(u) = u'/u here (chain would also be acceptable), so there is no wrong label or double step.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-12 — d/dx log(x-3) = 1/(x-3) is correct and the single step is labelled with a legitimate rule. SymPy proved every line and the final answer.qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 2 applies the derivative of the natural logarithm but fails to apply the chain rule for the inner function (x - 3). While the algebraic result is coincidentally correct because the derivative of the inner function is 1, the step claims to apply only the 'derivative' rule (unfolding d/dx on a known form) and ignores the composition, violating the single-rule-per-step constraint and mislabeling the operation.gpt-oss:20b: fail (style) 2026-09-26 — [domain objection, downgraded to style] Step 1 does not change the expression and is labeled with an undefined rule. It should either be omitted or replaced with a proper derivative step that applies the logarithm rule.qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 2 applies the derivative of the logarithm but fails to apply the chain rule for the inner function (x - 3). While the algebraic result is coincidentally correct because the derivative of (x - 3) is 1, the step claims to use only the 'derivative' rule, ignoring the composition structure. This is a defect because it skips the chain rule application, which is required by the 'one thing per step' contract when differentiating a composite function.gpt-oss:20b: fail (style) 2026-09-21 — Step 1 uses an unlabelled rule and does not actually apply any transformation; it should be labeled with an appropriate rule such as "rewrite" or omitted entirely.qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 2 applies the derivative of the natural logarithm but fails to apply the chain rule for the inner function (x - 3). While the algebraic result is coincidentally correct because the derivative of (x - 3) is 1, the step claims to apply only the 'derivative' rule for log, ignoring the necessary chain rule application required by the contract.gpt-oss:20b: fail (style) 2026-09-21 — [domain objection, downgraded to style] Step 1 does not change the expression and is labeled with an undefined rule. It should either be omitted or labeled as a rewrite/simplify step that merely restates the derivative operation.
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
gemma4:26b, checked 2026-09-26 with SymPy 1.14.0.