Derivative of \( \displaystyle \frac{x^{2}}{2} - x \left(x - 1\right) \ln{\left(2 x - 1 \right)} - \frac{x}{2} - \frac{\ln{\left(2 x - 1 \right)}}{4} \)
Problem 2.705 · hard
Differentiate \( \displaystyle f(x) = \frac{x^{2}}{2} - x \left(x - 1\right) \ln{\left(2 x - 1 \right)} - \frac{x}{2} - \frac{\ln{\left(2 x - 1 \right)}}{4} \).
- \[ \frac{d}{d x} \left(\frac{x^{2}}{2} - x \left(x - 1\right) \ln{\left(2 x - 1 \right)} - \frac{x}{2} - \frac{\ln{\left(2 x - 1 \right)}}{4}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{2} - \frac{d}{d x} x \left(x - 1\right) \ln{\left(2 x - 1 \right)} - \frac{d}{d x} \frac{\ln{\left(2 x - 1 \right)}}{4} \]sumApply the sum rule.✓ Proved
- \[ = - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{2} - \frac{d}{d x} x \left(x - 1\right) \ln{\left(2 x - 1 \right)} - \frac{\frac{d}{d x} \ln{\left(2 x - 1 \right)}}{4} \]constantFactor out the constant from the last term.✓ Proved
- \[ = - x \left(x - 1\right) \frac{d}{d x} \ln{\left(2 x - 1 \right)} - x \ln{\left(2 x - 1 \right)} \frac{d}{d x} \left(x - 1\right) - \left(x - 1\right) \ln{\left(2 x - 1 \right)} \frac{d}{d x} x - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{2} - \frac{\frac{d}{d x} \ln{\left(2 x - 1 \right)}}{4} \]productApply the product rule to the second term.✓ Proved
- \[ = - x \left(x - 1\right) \frac{d}{d x} \ln{\left(2 x - 1 \right)} - x \ln{\left(2 x - 1 \right)} - \left(x - 1\right) \ln{\left(2 x - 1 \right)} - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{2} - \frac{\frac{d}{d x} \ln{\left(2 x - 1 \right)}}{4} \]derivativeDifferentiate the individual parts of the product rule.✓ Proved
- \[ = - \frac{2 x \left(x - 1\right)}{2 x - 1} - x \ln{\left(2 x - 1 \right)} - \left(x - 1\right) \ln{\left(2 x - 1 \right)} - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{2} - \frac{\frac{d}{d x} \ln{\left(2 x - 1 \right)}}{4} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = - \frac{2 x \left(x - 1\right)}{2 x - 1} - x \ln{\left(2 x - 1 \right)} - \left(x - 1\right) \ln{\left(2 x - 1 \right)} - \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{x^{2}}{2} - \frac{1}{2 \left(2 x - 1\right)} \]algebraSimplify the derivative of the log term.✓ Proved
- \[ = - \frac{2 x \left(x - 1\right)}{2 x - 1} - x \ln{\left(2 x - 1 \right)} + x - \left(x - 1\right) \ln{\left(2 x - 1 \right)} - \frac{1}{2} - \frac{1}{4 x - 2} \]derivativeEvaluate the remaining derivatives.✓ Proved
- \[ = x - \left(2 x - 1\right) \ln{\left(2 x - 1 \right)} - \frac{1}{2} - \frac{1}{4 x - 2} - \frac{2 x^{2} - 2 x}{2 x - 1} \]algebra algebra algebraCombine the log terms. Group the rational terms.✓ Proved
- \[ = x - \frac{2 x}{4 x - 2} - \left(2 x - 1\right) \ln{\left(2 x - 1 \right)} - \frac{2 x^{2} - 2 x}{2 x - 1} \]algebra algebraFind a common denominator for the rational part. Simplify the numerator of the rational part.✓ Proved
- \[ = x - \frac{x}{2 x - 1} - \left(2 x - 1\right) \ln{\left(2 x - 1 \right)} - \frac{2 x^{2} - 2 x}{2 x - 1} \]algebraSimplify the fraction.✓ Proved
- \[ = x - \left(2 x - 1\right) \ln{\left(2 x - 1 \right)} - \frac{2 x^{2} - x}{2 x - 1} \]algebra algebraCombine the rational terms over the common denominator. Simplify the numerator.✓ Proved
- \[ = - \left(2 x - 1\right) \ln{\left(2 x - 1 \right)} \]algebra algebraFactor out x from the numerator. Simplify the fraction.✓ Proved
- \[ = \left(1 - 2 x\right) \ln{\left(2 x - 1 \right)} \]algebraFinal simplification.✓ Proved
Answer \( \left(1 - 2 x\right) \log{\left(2 x - 1 \right)} \)
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 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 undefined where 4*x - 2 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 2 = 0 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 4*x - 2 = 0 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 4*x - 2 = 0 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 4*x - 2 = 0 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 undefined where 4*x - 2 = 0 undefined where 2*x - 1 = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 2 = 0 undefined where 2*x - 1 = 0 |
| 15 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x - 1 = 0 |
| 16 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x - 1 = 0 |
| 17 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x - 1 = 0 |
| 18 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 19 | ✓ 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 4 applies the product rule twice in a single line, combining two separate rule applications into one. This violates the rule that each step must change only one thing. The correct approach would split the product rule application into two separate steps or use a single product rule on the two outer factors and then expand the inner product in a subsequent step.qwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: fail (error) 2026-09-21 — Step 4 applies the product rule twice in a single line, combining two separate rule applications into one. This violates the rule that each step must change only one thing. The correct approach would split the product rule application into two separate steps or use a single product rule on the two outer factors and then expand the inner product in a subsequent step.qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: pass 2026-09-21
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.