Derivative of \( \displaystyle - 2 x + \left(2 x + \frac{4}{3}\right) \ln{\left(3 x + 2 \right)} \)
Problem 2.1905 · hard
Differentiate \( \displaystyle f(x) = - 2 x + \left(2 x + \frac{4}{3}\right) \ln{\left(3 x + 2 \right)} \).
- \[ \frac{d}{d x} \left(- 2 x + \left(2 x + \frac{4}{3}\right) \ln{\left(3 x + 2 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- 2 x\right) + \frac{d}{d x} \left(2 x + \frac{4}{3}\right) \ln{\left(3 x + 2 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \left(- 2 x\right) + \frac{d}{d x} 2 x \ln{\left(3 x + 2 \right)} + \frac{d}{d x} \frac{4 \ln{\left(3 x + 2 \right)}}{3} \]productDistribute the logarithm term.✓ Proved
- \[ = \frac{d}{d x} \left(- 2 x\right) + \frac{d}{d x} 2 x \ln{\left(3 x + 2 \right)} + \frac{4 \frac{d}{d x} \ln{\left(3 x + 2 \right)}}{3} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = 2 x \frac{d}{d x} \ln{\left(3 x + 2 \right)} + \ln{\left(3 x + 2 \right)} \frac{d}{d x} 2 x + \frac{d}{d x} \left(- 2 x\right) + \frac{4 \frac{d}{d x} \ln{\left(3 x + 2 \right)}}{3} \]productApply the product rule to the second term.✓ Proved
- \[ = 2 x \frac{d}{d x} \ln{\left(3 x + 2 \right)} + 2 \ln{\left(3 x + 2 \right)} + \frac{d}{d x} \left(- 2 x\right) + \frac{4 \frac{d}{d x} \ln{\left(3 x + 2 \right)}}{3} \]derivativeDifferentiate the first part of the product rule.✓ Proved
- \[ = \left(2 x + \frac{4}{3}\right) \frac{d}{d x} \ln{\left(3 x + 2 \right)} + 2 \ln{\left(3 x + 2 \right)} + \frac{d}{d x} \left(- 2 x\right) \]algebraCombine the logarithmic derivative terms.✓ Proved
- \[ = \frac{\left(2 x + \frac{4}{3}\right) \frac{d}{d x} \left(3 x + 2\right)}{3 x + 2} + 2 \ln{\left(3 x + 2 \right)} + \frac{d}{d x} \left(- 2 x\right) \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \frac{3 \left(2 x + \frac{4}{3}\right)}{3 x + 2} + 2 \ln{\left(3 x + 2 \right)} + \frac{d}{d x} \left(- 2 x\right) \]derivativeDifferentiate the inner function 3x + 2.✓ Proved
- \[ = 2 \ln{\left(3 x + 2 \right)} + \frac{d}{d x} \left(- 2 x\right) + \frac{6 x + 4}{3 x + 2} \]algebraRearrange the terms.✓ Proved
- \[ = 2 \ln{\left(3 x + 2 \right)} - 2 + \frac{6 x + 4}{3 x + 2} \]derivative algebraDifferentiate the first term. Factor the numerator of the fraction.✓ Proved
- \[ = 2 \ln{\left(3 x + 2 \right)} \]simplify simplifyCancel the common term in the fraction. Combine the remaining constants.✓ Proved
Answer \( 2 \ln{\left(3 x + 2 \right)} \)
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 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 — The solution correctly applies differentiation rules in a step-by-step manner, adhering to the one-change-per-step constraint. All labels are appropriate for the operations performed.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly applies differentiation rules in a step-by-step manner, adhering to the one-change-per-step constraint. All labels are appropriate for the operations performed.qwen3.6:27b-mlx: fail (error) 2026-10-09 — Step 3 applies the distributive property (algebra) to split a product into a sum, but incorrectly labels it 'product'. The product rule is for differentiation, not for expanding algebraic terms.gpt-oss:20b: fail (style) 2026-10-09 — Step 3 incorrectly labels a distributive expansion as a "product" rule; it should be an "algebra" step. No other rule violations are present.
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 2026-10-09 with SymPy 1.14.0.