Derivative of \( \displaystyle \sqrt{2 x - 1} \ln{\left(2 x - 1 \right)} \)
Problem 2.621 · hard Video
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. How we verify
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: fail (error) — Step 4 applies both the power rule and the chain rule in one line, and step 7 applies both the logarithmic derivative and the chain rule together. Each step must change only one thing and label the appropriate single rule.qwen3.6:27b-mlx: pass — The solution correctly applies the product rule, chain rule, and algebraic simplifications in distinct steps. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.
Senior review claude-sonnet-5-5, 2026-10-11: pass — The solution is correct: each step is proved equal, labels come from the allowed vocabulary and name rules actually applied, and the final line matches the stated answer. Step 6 labelled constant-multiple for 1/2*2 is acceptable, and the algebra steps 10 and 11 are each a single simplification.
gpt-oss:20b: dismiss — Differentiating a power of an inner function is a chain-rule step, and evaluating the outer derivative (power or log) is part of that step per the contract; SymPy proved lines 4 and 7, and the labels 'chain' are legitimate.(a panel model that both passed and objected): dismiss — No specific defect was stated, and every line was proved equal by SymPy with legitimate rule labels.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-11 — The solution is correct: each step is proved equal, labels come from the allowed vocabulary and name rules actually applied, and the final line matches the stated answer. Step 6 labelled constant-multiple for 1/2*2 is acceptable, and the algebra steps 10 and 11 are each a single simplification.qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the product rule, chain rule, and algebraic simplifications in distinct steps. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.gpt-oss:20b: fail (error) 2026-09-26 — Step 4 applies both the power rule and the chain rule in one line, and step 7 applies both the logarithmic derivative and the chain rule together. Each step must change only one thing and label the appropriate single rule.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.