Derivative of \( \displaystyle - x^{2} + 2 x \left(x - 1\right) \ln{\left(2 x - 1 \right)} + x + \frac{\ln{\left(2 x - 1 \right)}}{2} \)
Problem 2.1358 · hard
Differentiate \( \displaystyle f(x) = - x^{2} + 2 x \left(x - 1\right) \ln{\left(2 x - 1 \right)} + x + \frac{\ln{\left(2 x - 1 \right)}}{2} \).
- \[ \frac{d}{d x} \left(- x^{2} + 2 x \left(x - 1\right) \ln{\left(2 x - 1 \right)} + x + \frac{\ln{\left(2 x - 1 \right)}}{2}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} x + \frac{d}{d x} \left(- x^{2}\right) + \frac{d}{d x} 2 x \left(x - 1\right) \ln{\left(2 x - 1 \right)} + \frac{d}{d x} \frac{\ln{\left(2 x - 1 \right)}}{2} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \left(- x^{2}\right) + \frac{d}{d x} 2 x \left(x - 1\right) \ln{\left(2 x - 1 \right)} + \frac{d}{d x} \frac{\ln{\left(2 x - 1 \right)}}{2} + 1 \]constantDifferentiate the term x.✓ Proved
- \[ = \frac{d}{d x} \left(- x^{2}\right) + \frac{d}{d x} 2 x \left(x - 1\right) \ln{\left(2 x - 1 \right)} + \frac{\frac{d}{d x} \ln{\left(2 x - 1 \right)}}{2} + 1 \]constant-multipleFactor out the constant 1/2.✓ Proved
- \[ = 2 x \left(x - 1\right) \frac{d}{d x} \ln{\left(2 x - 1 \right)} + \ln{\left(2 x - 1 \right)} \frac{d}{d x} 2 x \left(x - 1\right) + \frac{d}{d x} \left(- x^{2}\right) + \frac{\frac{d}{d x} \ln{\left(2 x - 1 \right)}}{2} + 1 \]productApply the product rule to the second term.✓ Proved
- \[ = 2 x \left(x - 1\right) \frac{d}{d x} \ln{\left(2 x - 1 \right)} + \ln{\left(2 x - 1 \right)} \frac{d}{d x} \left(2 x^{2} - 2 x\right) + \frac{d}{d x} \left(- x^{2}\right) + \frac{\frac{d}{d x} \ln{\left(2 x - 1 \right)}}{2} + 1 \]algebraExpand the product 2x(x-1).✓ Proved
- \[ = 2 x \left(x - 1\right) \frac{d}{d x} \ln{\left(2 x - 1 \right)} + \left(4 x - 2\right) \ln{\left(2 x - 1 \right)} + \frac{d}{d x} \left(- x^{2}\right) + \frac{\frac{d}{d x} \ln{\left(2 x - 1 \right)}}{2} + 1 \]derivativeDifferentiate the polynomial 2x^2 - 2x.✓ Proved
- \[ = \frac{4 x \left(x - 1\right)}{2 x - 1} + \left(4 x - 2\right) \ln{\left(2 x - 1 \right)} + \frac{d}{d x} \left(- x^{2}\right) + \frac{\frac{d}{d x} \ln{\left(2 x - 1 \right)}}{2} + 1 \]chainApply the chain rule to log(2x - 1).✓ Proved
- \[ = \frac{4 x \left(x - 1\right)}{2 x - 1} + \left(4 x - 2\right) \ln{\left(2 x - 1 \right)} + \frac{d}{d x} \left(- x^{2}\right) + 1 + \frac{1}{2 x - 1} \]derivativeDifferentiate the log term and simplify the fraction.✓ Proved
- \[ = - 2 x + \left(4 x - 2\right) \ln{\left(2 x - 1 \right)} + 1 + \frac{4 x^{2} - 4 x}{2 x - 1} + \frac{1}{2 x - 1} \]derivativeDifferentiate -x^2 and simplify constants.✓ Proved
- \[ = - 2 x + \left(4 x - 2\right) \ln{\left(2 x - 1 \right)} + 1 + \frac{4 x^{2} - 4 x + 1}{2 x - 1} \]algebraCombine the fractions with the same denominator.✓ Proved
- \[ = \left(4 x - 2\right) \ln{\left(2 x - 1 \right)} \]algebra simplify algebra simplifyRecognize the numerator as a perfect square. Simplify the fraction. Combine the linear terms. Final simplification.✓ Proved
Answer \( 2 \left(2 x - 1\right) \ln{\left(2 x - 1 \right)} \)
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 |
| 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 undefined where 2*x - 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x - 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x - 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x - 1 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x - 1 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 15 | ✓ 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 14 incorrectly combines the linear terms: -2*x + (2*x - 1) + 1 simplifies to 0, not 2*x. The subsequent simplification is therefore invalid.qwen3.6:27b-mlx: fail (style) — Step 9 applies the chain rule to the remaining log term and simplifies the fraction in a single step, violating the one-rule-per-step constraint. Step 10 applies the derivative to -x^2 and simplifies the rational expression in a single step, also violating the constraint.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-10-03 — Step 9 applies the chain rule to the remaining log term and simplifies the fraction in a single step, violating the one-rule-per-step constraint. Step 10 applies the derivative to -x^2 and simplifies the rational expression in a single step, also violating the constraint.gpt-oss:20b: fail (error) 2026-10-03 — Step 14 incorrectly combines the linear terms: -2*x + (2*x - 1) + 1 simplifies to 0, not 2*x. The subsequent simplification is therefore invalid.qwen3.6:27b-mlx: pass 2026-09-30 — The solution correctly applies differentiation rules and algebraic simplifications in a step-by-step manner. Each step adheres to the single-change constraint and uses valid labels from the provided vocabulary.gpt-oss:20b: fail (error) 2026-09-30 — Step 14 incorrectly combines the linear terms: -2*x + (2*x-1)+1 simplifies to 0, not +2*x. The correct simplification yields (4*x-2)*log(2*x-1) without any extra linear term.
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 2026-10-03 with SymPy 1.14.0.