Derivative of \( \displaystyle - x + \left(x + \frac{2}{3}\right) \ln{\left(3 x + 2 \right)} \)
Problem 2.1919 · hard Beautiful
Differentiate \( \displaystyle f(x) = - x + \left(x + \frac{2}{3}\right) \ln{\left(3 x + 2 \right)} \).
- \[ \frac{d}{d x} \left(- x + \left(x + \frac{2}{3}\right) \ln{\left(3 x + 2 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \left(x + \frac{2}{3}\right) \ln{\left(3 x + 2 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \left(x + \frac{2}{3}\right) \frac{d}{d x} \ln{\left(3 x + 2 \right)} + \ln{\left(3 x + 2 \right)} \frac{d}{d x} \left(x + \frac{2}{\operatorname{function}{\left(3 \right)}}\right) + \frac{d}{d x} \left(- x\right) \]productApply the product rule to the second term.✓ Proved
- \[ = \left(x + \frac{2}{3}\right) \frac{d}{d x} \ln{\left(3 x + 2 \right)} + \left(\frac{d}{d x} \frac{2}{3} + 1\right) \ln{\left(3 x + 2 \right)} + \frac{d}{d x} \left(- x\right) \]sumDistribute the derivative over the sum in the first part of the product rule.✓ Proved
- \[ = \left(x + \frac{2}{3}\right) \frac{d}{d x} \ln{\left(3 x + 2 \right)} + \ln{\left(3 x + 2 \right)} + \frac{d}{d x} \left(- x\right) \]constant algebraThe derivative of the constant 2/3 is 0. Simplify the term (1 + 0).✓ Proved
- \[ = \frac{\left(x + \frac{2}{3}\right) \frac{d}{d x} \left(3 x + 2\right)}{3 x + 2} + \ln{\left(3 x + 2 \right)} + \frac{d}{d x} \left(- x\right) \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \frac{3 \left(x + \frac{2}{3}\right)}{3 x + 2} + \ln{\left(3 x + 2 \right)} + \frac{d}{d x} \left(- x\right) \]sumDifferentiate the inner function 3*x + 2.✓ Proved
- \[ = \ln{\left(3 x + 2 \right)} + \frac{d}{d x} \left(- x\right) + 1 \]algebra algebraMultiply the constant 3 into the term (x + 2/3). Simplify the product of (3x + 2) and 1/(3x + 2).✓ Proved
- \[ = \ln{\left(3 x + 2 \right)} \]derivative simplifyEvaluate the derivative of -x. Combine the constant terms.✓ Proved
Answer \( \ln{\left(3 x + 2 \right)} \)
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. A reviewer's objection to the explanation stands; see 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 undefined where function(3) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where function(3) = 0 |
| 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 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 |
| 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 (error) — Step 3 applies two rules at once: it applies the product rule and incorrectly rewrites (x+2/3) as (x+2/function(3)), which is not a valid simplification. This violates the rule‑granularity requirement.qwen3.6:27b-mlx: fail (error) — Step 3 contains a syntax error ('2/function(3)') and incorrectly applies the product rule to a sum inside the first factor without separating the terms first. Step 4 incorrectly labels the distribution of the derivative over a sum as 'sum' while simultaneously performing algebraic simplification, violating the one-rule-per-step constraint.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-09 — Step 3 applies two rules at once: it applies the product rule and incorrectly rewrites (x+2/3) as (x+2/function(3)), which is not a valid simplification. This violates the rule‑granularity requirement.qwen3.6:27b-mlx: fail (error) 2026-10-09 — Step 3 contains a syntax error ('2/function(3)') and incorrectly applies the product rule to a sum inside the first factor without separating the terms first. Step 4 incorrectly labels the distribution of the derivative over a sum as 'sum' while simultaneously performing algebraic simplification, violating the one-rule-per-step constraint.gpt-oss:20b: fail (error) 2026-10-09 — Step 3 applies the product rule incorrectly and introduces a nonsensical term "Derivative(x + 2/function(3), x)". The correct application should be "Derivative(x + 2/3, x)*log(3*x + 2) + (x + 2/3)*Derivative(log(3*x + 2), x)". Additionally, step 9 incorrectly multiplies the constant 3 into the whole factor instead of just the derivative term, leading to an algebraic error.qwen3.6:27b-mlx: fail (error) 2026-10-09 — Step 3 contains a syntax error ('2/function(3)') and incorrectly applies the product rule to the first term of the sum in Step 2, violating the one-change-per-step rule by mixing sum and product operations. Step 8 is labeled 'sum' but performs differentiation of a linear term, which should be labeled 'derivative' or 'algebra' depending on interpretation, but the label 'sum' is incorrect for differentiation.
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.