Derivative of \( \displaystyle \sqrt{3 x - 1} \ln{\left(3 x - 1 \right)} \)
Problem 2.476 · hard
Differentiate \( \displaystyle f(x) = \sqrt{3 x - 1} \ln{\left(3 x - 1 \right)} \).
- \[ \frac{d}{d x} \sqrt{3 x - 1} \ln{\left(3 x - 1 \right)} \]Start with the derivative of the function.✓ Proved
- \[ = \sqrt{3 x - 1} \frac{d}{d x} \ln{\left(3 x - 1 \right)} + \ln{\left(3 x - 1 \right)} \frac{d}{d x} \sqrt{3 x - 1} \]product rewriteApply the product rule. Rewrite the square root as a power.✓ Proved
- \[ = \sqrt{3 x - 1} \frac{d}{d x} \ln{\left(3 x - 1 \right)} + \frac{\ln{\left(3 x - 1 \right)} \frac{d}{d x} \left(3 x - 1\right)}{2 \sqrt{3 x - 1}} \]powerApply the power rule to the first term.✓ Proved
- \[ = \sqrt{3 x - 1} \frac{d}{d x} \ln{\left(3 x - 1 \right)} + \frac{3 \ln{\left(3 x - 1 \right)}}{2 \sqrt{3 x - 1}} \]derivative constant-multipleDifferentiate the inner function 3*x - 1. Simplify the constant coefficient.✓ Proved
- \[ = \frac{3 \ln{\left(3 x - 1 \right)}}{2 \sqrt{3 x - 1}} + \frac{\frac{d}{d x} \left(3 x - 1\right)}{\sqrt{3 x - 1}} \]logarithmicApply the derivative rule for the logarithm.✓ Proved
- \[ = \frac{3 \ln{\left(3 x - 1 \right)}}{2 \sqrt{3 x - 1}} + \frac{3}{\sqrt{3 x - 1}} \]derivative constant-multiple algebraDifferentiate the inner function 3*x - 1 again. Simplify the second term. Simplify the power and the fraction.✓ Proved
- \[ = \frac{3 \ln{\left(3 x - 1 \right)} + 6}{2 \sqrt{3 x - 1}} \]simplifyCombine the terms over a common denominator.✓ Proved
Answer \( \frac{3 \left(\log{\left(3 x - 1 \right)} + 2\right)}{2 \sqrt{3 x - 1}} \)
✓ 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 3*x - 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x - 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x - 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x - 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x - 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x - 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x - 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*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 3*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
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the product rule, chain rule (via power and logarithmic rules), and algebraic simplification. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-20
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.