Derivative of \( \displaystyle 2 x \ln{\left(4 x - 1 \right)} - 2 x - \frac{\ln{\left(4 x - 1 \right)}}{2} \)
Problem 2.1326 · hard
Differentiate \( \displaystyle f(x) = 2 x \ln{\left(4 x - 1 \right)} - 2 x - \frac{\ln{\left(4 x - 1 \right)}}{2} \).
- \[ \frac{d}{d x} \left(2 x \ln{\left(4 x - 1 \right)} - 2 x - \frac{\ln{\left(4 x - 1 \right)}}{2}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} 2 x + \frac{d}{d x} 2 x \ln{\left(4 x - 1 \right)} - \frac{d}{d x} \frac{\ln{\left(4 x - 1 \right)}}{2} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} 2 x \ln{\left(4 x - 1 \right)} - \frac{d}{d x} \frac{\ln{\left(4 x - 1 \right)}}{2} - 2 \]constantDifferentiate the term 2*x.✓ Proved
- \[ = 2 x \frac{d}{d x} \ln{\left(4 x - 1 \right)} + \ln{\left(4 x - 1 \right)} \frac{d}{d x} 2 x - \frac{d}{d x} \frac{\ln{\left(4 x - 1 \right)}}{2} - 2 \]productApply the product rule to the first term.✓ Proved
- \[ = 2 x \frac{d}{d x} \ln{\left(4 x - 1 \right)} + 2 \ln{\left(4 x - 1 \right)} - \frac{d}{d x} \frac{\ln{\left(4 x - 1 \right)}}{2} - 2 \]derivativeDifferentiate 2*x.✓ Proved
- \[ = \frac{2 x \frac{d}{d x} \left(4 x - 1\right)}{4 x - 1} + 2 \ln{\left(4 x - 1 \right)} - \frac{d}{d x} \frac{\ln{\left(4 x - 1 \right)}}{2} - 2 \]chainApply the chain rule to log(4*x - 1).✓ Proved
- \[ = \frac{8 x}{4 x - 1} + 2 \ln{\left(4 x - 1 \right)} - \frac{d}{d x} \frac{\ln{\left(4 x - 1 \right)}}{2} - 2 \]derivative algebraDifferentiate 4*x - 1. Simplify the product.✓ Proved
- \[ = \frac{8 x}{4 x - 1} + 2 \ln{\left(4 x - 1 \right)} - \frac{\frac{d}{d x} \ln{\left(4 x - 1 \right)}}{2} - 2 \]constant-multipleFactor out 1/2 from the last term.✓ Proved
- \[ = \frac{8 x}{4 x - 1} + 2 \ln{\left(4 x - 1 \right)} - 2 - \frac{\frac{d}{d x} \left(4 x - 1\right)}{2 \left(4 x - 1\right)} \]chainApply the chain rule to the last term.✓ Proved
- \[ = \frac{8 x}{4 x - 1} + 2 \ln{\left(4 x - 1 \right)} - 2 - \frac{2}{4 x - 1} \]derivative algebraDifferentiate 4*x - 1. Simplify the last term.✓ Proved
- \[ = 2 \ln{\left(4 x - 1 \right)} - 2 + \frac{8 x - 2}{4 x - 1} \]algebraCombine the fractions.✓ Proved
- \[ = 2 \ln{\left(4 x - 1 \right)} \]algebra algebra simplify simplifyPut everything over a common denominator. Distribute the -2. Simplify the numerator. Final simplification.✓ Proved
Answer \( 2 \ln{\left(4 x - 1 \right)} \)
✓ 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 undefined where 4*x - 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*x - 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 4*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 - 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 - 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 - 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 - 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 - 1 = 0 |
| 15 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 16 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 17 | ✓ 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 step-by-step, adhering to the one-change-per-step constraint. All labels are appropriate for the operations performed.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. All labels are appropriate for the operations performed.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-09-30 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The algebraic simplification at the end is correct, and the labels used are appropriate for the operations performed.gpt-oss:20b: pass 2026-09-30
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.