Derivative of \( \displaystyle x - \left(x + \frac{2}{3}\right) \ln{\left(3 x + 2 \right)} \)
Problem 2.2006 · hard
Differentiate \( \displaystyle f(x) = x - \left(x + \frac{2}{3}\right) \ln{\left(3 x + 2 \right)} \).
- \[ \frac{d}{d x} \left(x - \left(x + \frac{2}{3}\right) \ln{\left(3 x + 2 \right)}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} x - \frac{d}{d x} \left(x + \frac{2}{3}\right) \ln{\left(3 x + 2 \right)} \]sumApply the difference rule.✓ Proved
- \[ = - \left(x + \frac{2}{3}\right) \frac{d}{d x} \ln{\left(3 x + 2 \right)} - \ln{\left(3 x + 2 \right)} \frac{d}{d x} \left(x + \frac{2}{3}\right) + \frac{d}{d x} x \]productApply the product rule to the second term.✓ Proved
- \[ = - \left(x + \frac{2}{3}\right) \frac{d}{d x} \ln{\left(3 x + 2 \right)} - \ln{\left(3 x + 2 \right)} \frac{d}{d x} \frac{2}{3} - \ln{\left(3 x + 2 \right)} \frac{d}{d x} x + \frac{d}{d x} x \]sumApply the sum rule to the first part of the product.✓ Proved
- \[ = - \left(x + \frac{2}{3}\right) \frac{d}{d x} \ln{\left(3 x + 2 \right)} - \ln{\left(3 x + 2 \right)} \frac{d}{d x} x + \frac{d}{d x} x \]constant simplifyThe derivative of the constant 2/3 is 0. Simplify the expression by removing the zero term.✓ Proved
- \[ = - \frac{\left(x + \frac{2}{3}\right) \frac{d}{d x} \left(3 x + 2\right)}{3 x + 2} - \ln{\left(3 x + 2 \right)} \frac{d}{d x} x + \frac{d}{d x} x \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = - \frac{3 \left(x + \frac{2}{3}\right) \frac{d}{d x} x}{3 x + 2} - \ln{\left(3 x + 2 \right)} \frac{d}{d x} x + \frac{d}{d x} x \]sumApply the sum rule to the argument of the logarithm.✓ Proved
- \[ = - \frac{3 \left(x + \frac{2}{3}\right)}{3 x + 2} - \ln{\left(3 x + 2 \right)} \frac{d}{d x} x + \frac{d}{d x} x \]constantThe derivative of x is 1.✓ Proved
- \[ = - \frac{3 \left(x + \frac{2}{3}\right)}{3 x + 2} - \ln{\left(3 x + 2 \right)} + 1 \]derivativeEvaluate the remaining derivatives.✓ Proved
- \[ = - \ln{\left(3 x + 2 \right)} \]algebra algebra simplify simplifyDistribute the negative sign. Distribute the 3 in the numerator. Simplify the fraction. Combine the constant terms.✓ Proved
Answer \( - \ln{\left(3 x + 2 \right)} \)
✓ 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.
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 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 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 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: fail (style) — Step 8 incorrectly labels the application of the constant‑multiple rule as a sum rule. The expression (1/(3*x+2))*3*Derivative(x,x) results from multiplying the constant 3 by the derivative of x, not from adding terms. This mislabeling could mislead a student about which rule applies.qwen3.6:27b-mlx: pass
Every verdict on record (4)
gpt-oss:20b: fail (style) 2026-10-10 — Step 8 incorrectly labels the application of the constant‑multiple rule as a sum rule. The expression (1/(3*x+2))*3*Derivative(x,x) results from multiplying the constant 3 by the derivative of x, not from adding terms. This mislabeling could mislead a student about which rule applies.qwen3.6:27b-mlx: pass 2026-10-10gpt-oss:20b: fail (style) 2026-10-10 — Step 8 applies the sum rule to a constant‑multiple expression; it should be labeled "constant-multiple" (or "product"), not "sum". This mislabeling violates the rule‑granularity requirement.qwen3.6:27b-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 transformations 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.