Derivative of \( \displaystyle 3 \ln{\left(x - 1 \right)} \)
Problem 2.955 · easy Mental math
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.
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: passqwen3.6:27b-mlx: fail (error) — 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.
Every verdict on record (4)
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 2026-09-27 with SymPy 1.14.0.