Derivative of \( \displaystyle \left(x - 1\right)^{2} \ln{\left(x - 1 \right)} \)
Problem 2.419 · hard
Differentiate \( \displaystyle f(x) = \left(x - 1\right)^{2} \ln{\left(x - 1 \right)} \).
- \[ \frac{d}{d x} \left(x - 1\right)^{2} \ln{\left(x - 1 \right)} \]Start with the derivative of the function.✓ Proved
- \[ = \left(x - 1\right)^{2} \frac{d}{d x} \ln{\left(x - 1 \right)} + \ln{\left(x - 1 \right)} \frac{d}{d x} \left(x - 1\right)^{2} \]productApply the product rule.✓ Proved
- \[ = x + \left(2 x - 2\right) \ln{\left(x - 1 \right)} - 1 \]chain algebraDifferentiate the individual parts using the chain rule. Simplify the first term.✓ Proved
- \[ = \left(x - 1\right) \left(2 \ln{\left(x - 1 \right)} + 1\right) \]algebraFactor out the common term (x - 1).✓ Proved
Answer \( \left(x - 1\right) \left(2 \log{\left(x - 1 \right)} + 1\right) \)
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 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: fail (error) — Step 3 applies two differentiation rules at once (chain for the log and power for the square) and mislabels the power derivative as chain. Each step must change only one thing and use the correct rule name.qwen3.6:27b-mlx: fail (style) — Step 3 applies two distinct differentiation rules (chain rule for the logarithm and power rule for the polynomial) simultaneously, violating the single-rule-per-step constraint. It should be split into two steps.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 3 applies two distinct differentiation rules (chain rule for the logarithm and power rule for the polynomial) simultaneously, violating the single-rule-per-step constraint. It should be split into two steps.gpt-oss:20b: fail (error) 2026-09-21 — Step 3 applies two differentiation rules at once (chain for the log and power for the square) and mislabels the power derivative as chain. Each step must change only one thing and use the correct rule name.qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 3 applies two distinct differentiation rules (the derivative of the logarithm and the power rule for the polynomial) simultaneously, violating the one-rule-per-step constraint. It should be split into two steps.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: fail (error) 2026-09-20 — Step 3 applies the chain rule to both terms in a single line, changing two expressions at once. Each step should modify only one part of the expression.
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.