Derivative of \( \displaystyle \ln{\left(x - 1 \right)} \)
Problem 2.335 · easy Mental math
Differentiate \( \displaystyle f(x) = \ln{\left(x - 1 \right)} \).
- \[ \frac{d}{d x} \ln{\left(x - 1 \right)} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{1}{x - 1} \]derivativeApply the derivative rule for the natural logarithm.✓ Proved
Answer \( \frac{1}{x - 1} \)
Mind the domain. The answer is also defined on (-oo, 1), where f(x) is not. Substituting there gives a number that is not a slope of f.
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. The reviewers disagree about how one step is explained; every verdict is in the receipt.
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 - 1 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where x - 1 = 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) — Step 1 does not change the expression; it merely restates the derivative. Each step must alter the expression by applying a single rule, so this step violates the contract.deepseek-r1:70b: fail (style) — The first step does not label the rule it applies, which is required by the contract.qwen3.6:27b-mlx: fail (error) — The step from log(x-1) to 1/(x-1) applies the chain rule implicitly but is labeled only as 'derivative'. Since the argument is not just x, the chain rule must be explicitly applied or labeled to satisfy the 'one rule per step' constraint.
Every verdict on record (13)
qwen3.6:27b-mlx: fail (error) 2026-09-20 — The step from log(x-1) to 1/(x-1) applies the chain rule implicitly but is labeled only as 'derivative'. Since the argument is not just x, the chain rule must be explicitly applied or labeled to satisfy the 'one rule per step' constraint.gpt-oss:20b: fail (style) 2026-09-20 — Step 1 does not change the expression; it merely restates the derivative. Each step must alter the expression by applying a single rule, so this step violates the contract.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 2 applies the derivative of the logarithm but fails to apply the chain rule for the inner function (x - 1), resulting in an incorrect label and missing factor. The step combines the outer derivative and the inner derivative into a single 'derivative' step, violating the one-rule-per-step constraint.gpt-oss:20b: fail (style) 2026-09-20 — [domain objection, downgraded to style] Step 1 does not change the expression and is labeled with an undefined rule. It should be omitted or replaced with a proper rule application.qwen3.6:27b-mlx: fail (error) 2026-09-19 — The step from line 1 to line 2 applies the chain rule to differentiate the composite function log(x-1), but is labeled 'derivative'. The label 'derivative' is reserved for basic forms (like d/dx log(x)), whereas this step requires 'chain' to account for the inner function (x-1). Applying a single rule label to a step that implicitly combines the logarithmic derivative rule and the chain rule violates the 'one rule per step' constraint.deepseek-r1:70b: fail (style) 2026-09-19 — The first step does not label the rule it applies, which is required by the contract.gpt-oss:20b: fail (error) 2026-09-19 — Step 2 applies two rules at once: the derivative of ln (u) and the chain rule for u=x−1. It should be split into a chain step (derivative of x−1) followed by a derivative step for ln. The label "derivative" alone is insufficient.qwen3.6:27b-mlx: fail (error) 2026-09-19 — The step from line 1 to line 2 applies the chain rule (differentiating the inner function x-1) but is labeled only as 'derivative'. This violates the single-rule-per-step constraint because the chain rule was not explicitly applied in a separate step.deepseek-r1:70b: fail (style) 2026-09-19 — First step lacks required rule label.gpt-oss:20b: fail (style) 2026-09-19 — Step 1 does not change the expression and is unlabeled; it should either be omitted or properly labeled as a rewrite or derivative application. The second step correctly applies the derivative rule for the natural logarithm.qwen3.6:27b-mlx: fail (error) 2026-09-18 — The step from line 1 to line 2 applies the chain rule (differentiating the inner function x-1) but is labeled only as 'derivative'. This violates the contract that each step must change only one thing and correctly name the rule applied; the chain rule was skipped.deepseek-r1:70b: fail (style) 2026-09-18 — First step should have rule label 'derivative'.gpt-oss:20b: fail (style) 2026-09-18 — Step 1 is labeled with an unlabelled rule and does not apply any transformation; it should be a rewrite or omitted. The second step correctly applies the derivative rule for ln(u).
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.