Derivative of \( \displaystyle \frac{9 x^{2} \left(2 \ln{\left(3 x \right)} - 1\right)}{4} \)
Problem 2.1067 · hard Video
Differentiate \( \displaystyle f(x) = \frac{9 x^{2} \left(2 \ln{\left(3 x \right)} - 1\right)}{4} \).
- \[ \frac{d}{d x} \frac{9 x^{2} \left(2 \ln{\left(3 x \right)} - 1\right)}{4} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{9 \frac{d}{d x} x^{2} \left(2 \ln{\left(3 x \right)} - 1\right)}{4} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{9 x^{2} \frac{d}{d x} \left(2 \ln{\left(3 x \right)} - 1\right)}{4} + \frac{9 \left(2 \ln{\left(3 x \right)} - 1\right) \frac{d}{d x} x^{2}}{4} \]productApply the product rule.✓ Proved
- \[ = \frac{9 x^{2} \frac{d}{d x} \left(2 \ln{\left(3 x \right)} - 1\right)}{4} + \frac{9 x \left(2 \ln{\left(3 x \right)} - 1\right)}{2} \]derivativeDifferentiate the first part of the product.✓ Proved
- \[ = \frac{9 x^{2} \left(- \frac{d}{d x} 1 + \frac{d}{d x} 2 \ln{\left(3 x \right)}\right)}{4} + \frac{9 x \left(2 \ln{\left(3 x \right)} - 1\right)}{2} \]sumDifferentiate the second part of the product.✓ Proved
- \[ = \frac{9 x^{2} \frac{d}{d x} \ln{\left(3 x \right)}}{2} + \frac{9 x \left(2 \ln{\left(3 x \right)} - 1\right)}{2} \]constantDifferentiate the constant term.✓ Proved
- \[ = \frac{9 x \left(2 \ln{\left(3 x \right)} - 1\right)}{2} + \frac{9 x}{2} \]chain algebra algebraApply the chain rule to the logarithm. Simplify the derivative of the logarithm. Simplify the term x**2 * (2/x).✓ Proved
- \[ = 9 x \ln{\left(3 x \right)} \]algebra simplify simplifyDistribute 2*x into the parentheses. Combine like terms. Final simplification.✓ Proved
Answer \( 9 x \ln{\left(3 x \right)} \)
✓ 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. 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 |
| 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 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 12 | ✓ 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 the constant multiple, product, and chain rules in separate steps. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies the constant multiple, product, and chain rules in separate steps. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. All labels are appropriate for the transformations performed.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies differentiation rules step-by-step, adhering to the single-change constraint and using valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-28
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 2026-09-29 with SymPy 1.14.0.