Derivative of \( \displaystyle - x \ln{\left(x - 1 \right)} + x + \ln{\left(x - 1 \right)} \)
Problem 2.1003 · hard Beautiful
Differentiate \( \displaystyle f(x) = - x \ln{\left(x - 1 \right)} + x + \ln{\left(x - 1 \right)} \).
- \[ \frac{d}{d x} \left(- x \ln{\left(x - 1 \right)} + x + \ln{\left(x - 1 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} x + \frac{d}{d x} \left(- x \ln{\left(x - 1 \right)}\right) + \frac{d}{d x} \ln{\left(x - 1 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \left(- x \ln{\left(x - 1 \right)}\right) + \frac{d}{d x} \ln{\left(x - 1 \right)} + 1 \]constantDifferentiate the term x.✓ Proved
- \[ = \frac{d}{d x} \left(- x \ln{\left(x - 1 \right)}\right) + 1 + \frac{1}{x - 1} \]logarithmicDifferentiate the log term.✓ Proved
- \[ = - x \frac{d}{d x} \ln{\left(x - 1 \right)} + \ln{\left(x - 1 \right)} \frac{d}{d x} \left(- x\right) + 1 + \frac{1}{x - 1} \]productApply the product rule to the first term.✓ Proved
- \[ = - \frac{x}{x - 1} - \ln{\left(x - 1 \right)} + 1 + \frac{1}{x - 1} \]derivative constant-multiple algebraDifferentiate the components of the product. Simplify the coefficient -1. Group the fractions together.✓ Proved
- \[ = \frac{1 - x}{x - 1} - \ln{\left(x - 1 \right)} + 1 \]algebraCombine the fractions.✓ Proved
- \[ = - \ln{\left(x - 1 \right)} \]algebra algebra simplifyRewrite 1 with a common denominator. Simplify the numerator. Final simplification.✓ Proved
Answer \( - \ln{\left(x - 1 \right)} \)
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 undefined where x - 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 12 | ✓ 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: passqwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and algebraic simplifications. Each step adheres to the single-change constraint and uses valid labels from the fixed vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies differentiation rules and algebraic simplifications. Each step adheres to the single-change constraint and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies differentiation rules and algebraic simplifications in single-step increments. All labels are appropriate for the operations performed.gpt-oss:20b: fail (error) 2026-09-27 — Step 6 applies two derivative rules in one step (both Derivative(-x) and Derivative(log(x-1))) while only one rule may be applied per step.
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.