Derivative of \( \displaystyle \frac{3 x^{2} \left(2 \ln{\left(3 x \right)} - 1\right)}{4} \)
Problem 2.1018 · hard
Differentiate \( \displaystyle f(x) = \frac{3 x^{2} \left(2 \ln{\left(3 x \right)} - 1\right)}{4} \).
- \[ \frac{d}{d x} \frac{3 x^{2} \left(2 \ln{\left(3 x \right)} - 1\right)}{4} \]constant-multiplePull out the constant coefficient.✓ Proved
- \[ = \frac{3 \frac{d}{d x} x^{2} \left(2 \ln{\left(3 x \right)} - 1\right)}{4} \]productApply the product rule.✓ Proved
- \[ = \frac{3 x^{2} \frac{d}{d x} \left(2 \ln{\left(3 x \right)} - 1\right)}{4} + \frac{3 \left(2 \ln{\left(3 x \right)} - 1\right) \frac{d}{d x} x^{2}}{4} \]derivativeDifferentiate the first part of the product.✓ Proved
- \[ = \frac{3 x^{2} \frac{d}{d x} \left(2 \ln{\left(3 x \right)} - 1\right)}{4} + \frac{3 x \left(2 \ln{\left(3 x \right)} - 1\right)}{2} \]sumDifferentiate the terms inside the second part of the product.✓ Proved
- \[ = \frac{3 x^{2} \left(- \frac{d}{d x} 1 + \frac{d}{d x} 2 \ln{\left(3 x \right)}\right)}{4} + \frac{3 x \left(2 \ln{\left(3 x \right)} - 1\right)}{2} \]derivativeDifferentiate the terms inside the parentheses.✓ Proved
- \[ = \frac{3 x^{2} \frac{d}{d x} \ln{\left(3 x \right)}}{2} + \frac{3 x \left(2 \ln{\left(3 x \right)} - 1\right)}{2} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \frac{3 x \left(2 \ln{\left(3 x \right)} - 1\right)}{2} + \frac{3 x}{2} \]algebra algebra algebraSimplify the derivative of the log term. Simplify the expression inside the parentheses. Distribute x**2 into the second term.✓ Proved
- \[ = 3 x \ln{\left(3 x \right)} \]algebra algebra simplifyDistribute 2*x into the parentheses. Combine like terms. Final simplification.✓ Proved
Answer \( 3 x \ln{\left(3 x \right)} \)
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. The reviewers disagree about how one step is explained; every verdict is in the receipt.
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: fail (style) — Step 4 incorrectly labels a derivative as a sum; step 5 incorrectly labels a sum as a derivative. Each step must change only one rule, and the rule name must match the operation performed.qwen3.6:27b-mlx: fail (style) — Step 3 is labeled 'derivative' but applies the product rule expansion; it should be labeled 'product'. Step 4 is labeled 'sum' but performs the differentiation of x^2; it should be labeled 'derivative'. The labels are swapped or misapplied relative to the actions taken.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-09-27 — Step 3 is labeled 'derivative' but applies the product rule expansion; it should be labeled 'product'. Step 4 is labeled 'sum' but performs the differentiation of x^2; it should be labeled 'derivative'. The labels are swapped or misapplied relative to the actions taken.gpt-oss:20b: fail (style) 2026-09-27 — Step 4 incorrectly labels a derivative as a sum; step 5 incorrectly labels a sum as a derivative. Each step must change only one rule, and the rule name must match the operation performed.qwen3.6:27b-mlx: fail (error) 2026-09-27 — Step 3 is labeled 'derivative' but performs the product rule expansion (u'v + uv'), which is a multi-rule operation not covered by the 'derivative' label (reserved for basic forms). Additionally, Step 4 is labeled 'sum' but performs differentiation of the second term, which is a differentiation step, not a sum rule application.gpt-oss:20b: fail (style) 2026-09-27 — Step 4 applies a derivative (d/dx of x**2) but labels it as "sum". The correct label should be "derivative"; no other rule is applied in that step.
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 2026-09-27 with SymPy 1.14.0.