Derivative of \( \displaystyle 2 x \ln{\left(3 x + 2 \right)} - 2 x + \frac{4 \ln{\left(3 x + 2 \right)}}{3} \)
Problem 2.894 · hard Video
Differentiate \( \displaystyle f(x) = 2 x \ln{\left(3 x + 2 \right)} - 2 x + \frac{4 \ln{\left(3 x + 2 \right)}}{3} \).
- \[ \frac{d}{d x} \left(2 x \ln{\left(3 x + 2 \right)} - 2 x + \frac{4 \ln{\left(3 x + 2 \right)}}{3}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} 2 x + \frac{d}{d x} 2 x \ln{\left(3 x + 2 \right)} + \frac{d}{d x} \frac{4 \ln{\left(3 x + 2 \right)}}{3} \]sumApply the sum rule for differentiation.✓ Proved
- \[ = - \frac{d}{d x} 2 x + \frac{d}{d x} 2 x \ln{\left(3 x + 2 \right)} + \frac{4 \frac{d}{d x} \ln{\left(3 x + 2 \right)}}{3} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = - \frac{d}{d x} 2 x + \frac{d}{d x} 2 x \ln{\left(3 x + 2 \right)} + \frac{4 \frac{d}{d x} \left(3 x + 2\right)}{3 \left(3 x + 2\right)} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = - \frac{d}{d x} 2 x + \frac{d}{d x} 2 x \ln{\left(3 x + 2 \right)} + \frac{4}{3 x + 2} \]derivative algebraDifferentiate the inner function 3*x + 2. Simplify the constant terms.✓ Proved
- \[ = 2 x \frac{d}{d x} \ln{\left(3 x + 2 \right)} + \ln{\left(3 x + 2 \right)} \frac{d}{d x} 2 x - \frac{d}{d x} 2 x + \frac{4}{3 x + 2} \]productApply the product rule to the first term.✓ Proved
- \[ = \frac{2 x \frac{d}{d x} \left(3 x + 2\right)}{3 x + 2} + 2 \ln{\left(3 x + 2 \right)} - 2 + \frac{4}{3 x + 2} \]derivativeDifferentiate the components of the product rule.✓ Proved
- \[ = \frac{6 x}{3 x + 2} + 2 \ln{\left(3 x + 2 \right)} - 2 + \frac{4}{3 x + 2} \]derivative algebraDifferentiate the inner function 3*x + 2. Multiply the terms in the second part.✓ Proved
- \[ = 2 \ln{\left(3 x + 2 \right)} - 2 + \frac{6 x + 4}{3 x + 2} \]algebra algebraCombine the fractions with a common denominator. Factor the numerator of the fraction.✓ Proved
- \[ = 2 \ln{\left(3 x + 2 \right)} \]simplify simplifyCancel the common term in the fraction. Final simplification.✓ Proved
Answer \( 2 \ln{\left(3 x + 2 \right)} \)
✓ 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 undefined where 3*x + 2 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 2 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 2 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 2 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 2 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 2 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 2 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 2 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 2 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 2 = 0 |
| 14 | ✓ 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
Senior review claude-sonnet-5-5, 2026-10-13: pass — The derivative is correct: the log terms and constants combine as 2 log(3x+2) + (6x+4)/(3x+2) - 2 = 2 log(3x+2). Line 8 evaluates several simple derivatives at once under the label 'derivative', which is acceptable and at most a style quibble.
panel (unnamed model that objected): dismiss — No specific defect was stated. Every line checked equal under SymPy, and the rule labels match the steps, so there is nothing to uphold.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-13 — The derivative is correct: the log terms and constants combine as 2 log(3x+2) + (6x+4)/(3x+2) - 2 = 2 log(3x+2). Line 8 evaluates several simple derivatives at once under the label 'derivative', which is acceptable and at most a style quibble.qwen3.6:27b-mlx: pass 2026-09-27gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies differentiation rules one at a time and uses valid labels from the fixed vocabulary. The algebraic simplifications are sound and clearly separated from the differentiation steps.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-26gpt-oss:20b: fail (error) 2026-09-26 — Step 8 applies two derivative rules in one line (derivative of 2*x and derivative of log(3*x+2)), violating the single‑rule‑change requirement.
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.