Derivative of \( \displaystyle \sqrt{2 x - 1} \ln{\left(2 x - 1 \right)} \)
Problem 2.621 · hard
Differentiate \( \displaystyle f(x) = \sqrt{2 x - 1} \ln{\left(2 x - 1 \right)} \).
- \[ \frac{d}{d x} \sqrt{2 x - 1} \ln{\left(2 x - 1 \right)} \]Start with the derivative of the function.✓ Proved
- \[ = \sqrt{2 x - 1} \frac{d}{d x} \ln{\left(2 x - 1 \right)} + \ln{\left(2 x - 1 \right)} \frac{d}{d x} \sqrt{2 x - 1} \]product rewriteApply the product rule. Rewrite the square root as a power.✓ Proved
- \[ = \sqrt{2 x - 1} \frac{d}{d x} \ln{\left(2 x - 1 \right)} + \frac{\ln{\left(2 x - 1 \right)} \frac{d}{d x} \left(2 x - 1\right)}{2 \sqrt{2 x - 1}} \]chainApply the chain rule to the first term.✓ Proved
- \[ = \sqrt{2 x - 1} \frac{d}{d x} \ln{\left(2 x - 1 \right)} + \frac{\ln{\left(2 x - 1 \right)}}{\sqrt{2 x - 1}} \]derivative constant-multipleDifferentiate the inner function 2*x - 1. Simplify the constant factor.✓ Proved
- \[ = \frac{\ln{\left(2 x - 1 \right)}}{\sqrt{2 x - 1}} + \frac{\frac{d}{d x} \left(2 x - 1\right)}{\sqrt{2 x - 1}} \]chainApply the chain rule to the second term.✓ Proved
- \[ = \frac{\ln{\left(2 x - 1 \right)}}{\sqrt{2 x - 1}} + \frac{2}{\sqrt{2 x - 1}} \]derivative constant-multiple algebraDifferentiate the inner function 2*x - 1. Simplify the constant factor. Simplify the power and the fraction.✓ Proved
- \[ = \frac{\ln{\left(2 x - 1 \right)} + 2}{\sqrt{2 x - 1}} \]algebraCombine the terms over a common denominator.✓ Proved
Answer \( \frac{\log{\left(2 x - 1 \right)} + 2}{\sqrt{2 x - 1}} \)
✓ 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 undefined where 2*x - 1 = 0 |
| 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 |
| 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 |
| 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 undefined where 2*x - 1 = 0 |
| 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 the product rule, chain rule, and algebraic simplifications in distinct steps. All labels accurately reflect the operations performed.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the product rule, chain rule, and algebraic simplifications in distinct steps. All labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the product rule, chain rule, and algebraic simplifications in distinct steps. All labels are appropriate and the logic is sound.gpt-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.