Derivative of \( \displaystyle - \frac{x^{2}}{2} + \frac{x}{2} + \left(x^{2} - x + \frac{1}{4}\right) \ln{\left(2 x - 1 \right)} \)
Problem 2.1684 · hard
Differentiate \( \displaystyle f(x) = - \frac{x^{2}}{2} + \frac{x}{2} + \left(x^{2} - x + \frac{1}{4}\right) \ln{\left(2 x - 1 \right)} \).
- \[ \frac{d}{d x} \left(- \frac{x^{2}}{2} + \frac{x}{2} + \left(x^{2} - x + \frac{1}{4}\right) \ln{\left(2 x - 1 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{d}{d x} \left(x^{2} - x + \frac{1}{4}\right) \ln{\left(2 x - 1 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \left(x^{2} - x + \frac{1}{4}\right) \frac{d}{d x} \ln{\left(2 x - 1 \right)} + \ln{\left(2 x - 1 \right)} \frac{d}{d x} \left(x^{2} - x + \frac{1}{4}\right) + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) \]productApply the product rule to the third term.✓ Proved
- \[ = \left(2 x - 1\right) \ln{\left(2 x - 1 \right)} + \left(x^{2} - x + \frac{1}{4}\right) \frac{d}{d x} \ln{\left(2 x - 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) \]derivativeDifferentiate the polynomial part.✓ Proved
- \[ = \left(2 x - 1\right) \ln{\left(2 x - 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{\left(x^{2} - x + \frac{1}{4}\right) \frac{d}{d x} \left(2 x - 1\right)}{2 x - 1} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \left(2 x - 1\right) \ln{\left(2 x - 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{2 \left(x^{2} - x + \frac{1}{4}\right)}{2 x - 1} \]derivativeDifferentiate the inner function of the logarithm.✓ Proved
- \[ = \left(2 x - 1\right) \ln{\left(2 x - 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \left(- \frac{x^{2}}{2}\right) + \frac{2 x^{2} - 2 x + \frac{1}{2}}{2 x - 1} \]algebraSimplify the fraction.✓ Proved
- \[ = - x + \left(2 x - 1\right) \ln{\left(2 x - 1 \right)} + \frac{1}{2} + \frac{2 x^{2} - 2 x + \frac{1}{2}}{2 x - 1} \]derivative simplifyEvaluate the derivatives of the first two terms. Final simplification.✓ Proved
Answer \( \left(2 x - 1\right) \ln{\left(2 x - 1 \right)} \)
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 undefined where 2*x - 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x - 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x - 1 = 0 |
| 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 |
| 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 step-by-step, adhering to the one-change-per-step constraint. The algebraic simplification in the final steps is correct, as (2x^2 - 2x + 1/2) / (2x - 1) simplifies to x - 1/2, which cancels with the -x + 1/2 terms, leaving the stated answer.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The algebraic simplification in the final steps is correct, as (2x^2 - 2x + 1/2) / (2x - 1) simplifies to x - 1/2, which cancels with the -x + 1/2 terms, leaving the stated answer.gpt-oss:20b: fail (error) 2026-10-06 — The solution fails to simplify the rational term (2*x**2-2*x+1/2)/(2*x-1) to x-1/2, so the final expression still contains an extraneous term. The correct derivative is (2*x-1)*log(2*x-1).qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies the sum, product, and chain rules in separate steps. The algebraic simplification in step 7 is correct, and the final evaluation of derivatives is sound.
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 2026-10-06 with SymPy 1.14.0.