Derivative of \( \displaystyle - x + \left(x - 1\right) \ln{\left(x - 1 \right)} \)
Problem 2.1487 · hard Beautiful
Differentiate \( \displaystyle f(x) = - x + \left(x - 1\right) \ln{\left(x - 1 \right)} \).
- \[ \frac{d}{d x} \left(- x + \left(x - 1\right) \ln{\left(x - 1 \right)}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \left(x - 1\right) \ln{\left(x - 1 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \left(x - 1\right) \frac{d}{d x} \ln{\left(x - 1 \right)} + \ln{\left(x - 1 \right)} \frac{d}{d x} \left(x - 1\right) + \frac{d}{d x} \left(- x\right) \]productApply the product rule to the second term.✓ Proved
- \[ = \left(x - 1\right) \frac{d}{d x} \ln{\left(x - 1 \right)} + \ln{\left(x - 1 \right)} \frac{d}{d x} \left(x - 1\right) - 1 \]constantDifferentiate the first term.✓ Proved
- \[ = \left(x - 1\right) \frac{d}{d x} \ln{\left(x - 1 \right)} + \ln{\left(x - 1 \right)} - 1 \]derivativeDifferentiate the term (x - 1).✓ Proved
- \[ = \ln{\left(x - 1 \right)} \]derivative algebra simplifyDifferentiate the log term. Simplify the fraction. Combine the constant terms.✓ Proved
Answer \( \ln{\left(x - 1 \right)} \)
✓ 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.
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 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 8 | ✓ 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 (style) — Step 4 incorrectly labels the differentiation of -x as "constant"; it should be labeled "derivative". This mislabeling could mislead a student about which rule applies to a linear term.qwen3.6:27b-mlx: pass — The solution correctly applies the sum, product, and chain rules (via the derivative rule for composite functions) in a step-by-step manner. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the sum, product, and chain rules (via the derivative rule for composite functions) in a step-by-step manner. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.gpt-oss:20b: fail (style) 2026-10-04 — Step 4 incorrectly labels the differentiation of -x as "constant"; it should be labeled "derivative". This mislabeling could mislead a student about which rule applies to a linear term.qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the sum, product, and chain rules (via the derivative rule for log) in separate steps. Each step changes only one aspect of the expression, and the labels are appropriate.gpt-oss:20b: pass 2026-10-04
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 2026-10-04 with SymPy 1.14.0.