Derivative of \( \displaystyle 3 \ln{\left(x - 1 \right)} \)
Problem 2.955 · easy Mental math Video
Differentiate \( \displaystyle f(x) = 3 \ln{\left(x - 1 \right)} \).
- \[ \frac{d}{d x} 3 \ln{\left(x - 1 \right)} \]Start with the derivative of the function.✓ Proved
- \[ = 3 \frac{d}{d x} \ln{\left(x - 1 \right)} \]constant-multipleMove the constant outside the derivative.✓ Proved
- \[ = \frac{3}{x - 1} \]derivative simplifyApply the derivative rule for the natural logarithm. Simplify the expression.✓ Proved
Answer \( \frac{3}{x - 1} \)
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. 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 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 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 lacks a proper rule label from the allowed vocabulary; every step must name a rule such as derivative, constant-multiple, etc.qwen3.6:27b-mlx: fail (error) — Step 3 applies the derivative of the logarithm but fails to apply the chain rule for the inner function (x - 1). While the derivative of (x-1) is 1, making the final result numerically correct, the step skips a required rule application, violating the 'one rule per step' contract.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — Step 3 applies the derivative of the logarithm but fails to apply the chain rule for the inner function (x - 1). While the derivative of (x-1) is 1, making the final result numerically correct, the step skips a required rule application, violating the 'one rule per step' contract.gpt-oss:20b: fail (style) 2026-09-28 — Step 1 lacks a proper rule label from the allowed vocabulary; every step must name a rule such as derivative, constant-multiple, etc.qwen3.6:27b-mlx: fail (error) 2026-09-27 — Step 3 is mathematically incorrect because it applies the derivative of log(u) as 1/u, omitting the necessary chain rule factor 1/u' (which is 1 in this specific case, but the step label 'derivative' does not account for the inner function differentiation, and the step itself skips the chain rule application entirely). A correct derivation would require a step applying the chain rule to log(x-1) to get 1/(x-1) * 1, or the step should be labeled 'chain' if it implicitly handles it, but the note claims it applies the 'derivative rule for the natural logarithm' which is incomplete for composite functions.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-27 — Step 3 applies the derivative of log(u) as 1/u, omitting the chain rule factor u'. The derivative of log(x-1) is 1/(x-1) * 1, but the step label 'derivative' implies a direct application without accounting for the inner function's derivative, which is a missing logical step or mislabeling of the chain rule application.gpt-oss:20b: fail (error) 2026-09-27 — Step 3 applies the derivative of log(x-1) but ignores the chain rule; the derivative should be 1/(x-1)·1, so a chain rule step is missing. Step 1 is also unlabeled.
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 2026-09-28 with SymPy 1.14.0.