Derivative of \( \displaystyle x + \left(\frac{1}{3} - x\right) \ln{\left(3 x - 1 \right)} \)
Problem 2.1917 · hard
Differentiate \( \displaystyle f(x) = x + \left(\frac{1}{3} - x\right) \ln{\left(3 x - 1 \right)} \).
- \[ \frac{d}{d x} \left(x + \left(\frac{1}{3} - x\right) \ln{\left(3 x - 1 \right)}\right) \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} x + \frac{d}{d x} \left(\frac{1}{3} - x\right) \ln{\left(3 x - 1 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \left(\frac{1}{3} - x\right) \ln{\left(3 x - 1 \right)} + 1 \]constantDifferentiate the first term.✓ Proved
- \[ = \left(\frac{1}{3} - x\right) \frac{d}{d x} \ln{\left(3 x - 1 \right)} + \ln{\left(3 x - 1 \right)} \frac{d}{d x} \left(\frac{1}{3} - x\right) + 1 \]productApply the product rule to the second term.✓ Proved
- \[ = \left(\frac{1}{3} - x\right) \frac{d}{d x} \ln{\left(3 x - 1 \right)} - \ln{\left(3 x - 1 \right)} + 1 \]derivativeDifferentiate the first part of the product.✓ Proved
- \[ = \frac{\left(\frac{1}{3} - x\right) \frac{d}{d x} \left(3 x - 1\right)}{3 x - 1} - \ln{\left(3 x - 1 \right)} + 1 \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \frac{3 \left(\frac{1}{3} - x\right)}{3 x - 1} - \ln{\left(3 x - 1 \right)} + 1 \]derivativeDifferentiate the inner function of the log.✓ Proved
- \[ = \frac{1 - 3 x}{3 x - 1} - \ln{\left(3 x - 1 \right)} + 1 \]algebra algebraSimplify the expression. Distribute the 3 in the numerator.✓ Proved
- \[ = - \ln{\left(3 x - 1 \right)} \]algebra simplify simplifyFactor out a negative sign from the numerator. Simplify the fraction. Combine the constant terms.✓ Proved
Answer \( - \ln{\left(3 x - 1 \right)} \)
✓ 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 |
| 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 |
| 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: passqwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and algebraic simplifications. Each step changes only one aspect of the expression, and the labels accurately reflect 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 and algebraic simplifications. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels used 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 2026-10-09 with SymPy 1.14.0.