Derivative of \( \displaystyle - \frac{3 x^{2}}{2} - x + \left(3 x^{2} + 2 x + \frac{1}{3}\right) \ln{\left(3 x + 1 \right)} \)
Problem 2.2012 · hard
Differentiate \( \displaystyle f(x) = - \frac{3 x^{2}}{2} - x + \left(3 x^{2} + 2 x + \frac{1}{3}\right) \ln{\left(3 x + 1 \right)} \).
- \[ \frac{d}{d x} \left(- \frac{3 x^{2}}{2} - x + \left(3 x^{2} + 2 x + \frac{1}{3}\right) \ln{\left(3 x + 1 \right)}\right) \]derivativeDifferentiate the entire function.✓ Proved
- \[ = - \frac{d}{d x} x + \frac{d}{d x} \left(- \frac{3 x^{2}}{2}\right) + \frac{d}{d x} \left(3 x^{2} + 2 x + \frac{1}{3}\right) \ln{\left(3 x + 1 \right)} \]sumApply the sum rule to separate the terms.✓ Proved
- \[ = \left(3 x^{2} + 2 x + \frac{1}{3}\right) \frac{d}{d x} \ln{\left(3 x + 1 \right)} + \ln{\left(3 x + 1 \right)} \frac{d}{d x} \left(3 x^{2} + 2 x + \frac{1}{3}\right) - \frac{d}{d x} x + \frac{d}{d x} \left(- \frac{3 x^{2}}{2}\right) \]productApply the product rule to the third term.✓ Proved
- \[ = \left(6 x + 2\right) \ln{\left(3 x + 1 \right)} + \left(3 x^{2} + 2 x + \frac{1}{3}\right) \frac{d}{d x} \ln{\left(3 x + 1 \right)} - \frac{d}{d x} x + \frac{d}{d x} \left(- \frac{3 x^{2}}{2}\right) \]derivativeDifferentiate the polynomial part.✓ Proved
- \[ = \left(6 x + 2\right) \ln{\left(3 x + 1 \right)} - \frac{d}{d x} x + \frac{d}{d x} \left(- \frac{3 x^{2}}{2}\right) + \frac{\left(3 x^{2} + 2 x + \frac{1}{3}\right) \frac{d}{d x} \left(3 x + 1\right)}{3 x + 1} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \left(6 x + 2\right) \ln{\left(3 x + 1 \right)} - \frac{d}{d x} x + \frac{d}{d x} \left(- \frac{3 x^{2}}{2}\right) + \frac{3 \left(3 x^{2} + 2 x + \frac{1}{3}\right)}{3 x + 1} \]derivativeDifferentiate the inner function 3*x + 1.✓ Proved
- \[ = \left(6 x + 2\right) \ln{\left(3 x + 1 \right)} - \frac{d}{d x} x + \frac{d}{d x} \left(- \frac{3 x^{2}}{2}\right) + \frac{9 x^{2} + 6 x + 1}{3 x + 1} \]algebraSimplify the multiplication and fraction.✓ Proved
- \[ = \left(6 x + 2\right) \ln{\left(3 x + 1 \right)} + \frac{d}{d x} \left(- \frac{3 x^{2}}{2}\right) - 1 + \frac{9 x^{2} + 6 x + 1}{3 x + 1} \]derivativeDifferentiate the term x.✓ Proved
- \[ = - 3 x + \left(6 x + 2\right) \ln{\left(3 x + 1 \right)} - 1 + \frac{9 x^{2} + 6 x + 1}{3 x + 1} \]derivative algebraDifferentiate the term -3*x**2/2. Simplify the first term.✓ Proved
- \[ = \left(6 x + 2\right) \ln{\left(3 x + 1 \right)} \]algebra simplify simplifyRecognize the numerator as a perfect square. Simplify the fraction and combine terms. The remaining terms cancel out.✓ Proved
Answer \( 2 \left(3 x + 1\right) \ln{\left(3 x + 1 \right)} \)
✓ 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.
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 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 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 13 | ✓ 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 differentiation rules step-by-step, adhering to the one-change-per-step constraint. The algebraic simplifications in the final steps are valid and lead to the correct result.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The algebraic simplifications in the final steps are valid and lead to the correct result.gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. All labels are appropriate for the operations performed.
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 2026-10-10 with SymPy 1.14.0.