Home›Calculus 1›Derivatives of logarithms›Problem 2.419

Derivative of \( \displaystyle \left(x - 1\right)^{2} \ln{\left(x - 1 \right)} \)

Problem 2.419 · hard Video

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 \( \left(x - 1\right) \left(2 \log{\left(x - 1 \right)} + 1\right) \)

Top: the graph of f = (x - 1)²·log(x - 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = x + (2x - 2)·log(x - 1) - 1.

✓ 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
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.

Senior review claude-sonnet-5-5, 2026-10-08: pass — The product rule is applied correctly, the derivatives of the parts are right, and the algebra and factoring steps are valid. SymPy proved every line, including the final line against the stated answer.

gpt-oss:20b: dismiss — Both pieces are chain-rule evaluations: log(x-1) has inner function x-1, and (x-1)**2 is a power of the inner function x-1, so 'chain' names a rule that was actually applied and is not wrong.
qwen3.6:27b-mlx: dismiss — Step 3 only evaluates the two pending Derivative placeholders left by the product rule, which is one change, and a label that fits either term is acceptable under the contract.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-08 — The product rule is applied correctly, the derivatives of the parts are right, and the algebra and factoring steps are valid. SymPy proved every line, including the final line against the stated answer.
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 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-20
gpt-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.