Derivative of \( \displaystyle - 2 x + \left(2 x + \frac{2}{3}\right) \ln{\left(3 x + 1 \right)} \)
Problem 2.1987 · hard
Differentiate \( \displaystyle f(x) = - 2 x + \left(2 x + \frac{2}{3}\right) \ln{\left(3 x + 1 \right)} \).
- \[ \frac{d}{d x} \left(- 2 x + \left(2 x + \frac{2}{3}\right) \ln{\left(3 x + 1 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- 2 x\right) + \frac{d}{d x} \left(2 x + \frac{2}{3}\right) \ln{\left(3 x + 1 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \left(- 2 x\right) + \frac{d}{d x} 2 x \ln{\left(3 x + 1 \right)} + \frac{d}{d x} \frac{2 \ln{\left(3 x + 1 \right)}}{3} \]productApply the product rule to the second term.✓ Proved
- \[ = \frac{d}{d x} \left(- 2 x\right) + \frac{d}{d x} 2 x \ln{\left(3 x + 1 \right)} + \frac{2 \frac{d}{d x} \ln{\left(3 x + 1 \right)}}{3} \]constant-multiplePull out the constant from the third term.✓ Proved
- \[ = 2 x \frac{d}{d x} \ln{\left(3 x + 1 \right)} + \ln{\left(3 x + 1 \right)} \frac{d}{d x} 2 x + \frac{d}{d x} \left(- 2 x\right) + \frac{2 \frac{d}{d x} \ln{\left(3 x + 1 \right)}}{3} \]productApply the product rule to the second term.✓ Proved
- \[ = 2 x \frac{d}{d x} \ln{\left(3 x + 1 \right)} + 2 \ln{\left(3 x + 1 \right)} + \frac{2 \frac{d}{d x} \ln{\left(3 x + 1 \right)}}{3} - 2 \]derivative simplifyDifferentiate the individual terms. Remove unnecessary parentheses.✓ Proved
- \[ = \left(2 x + \frac{2}{3}\right) \frac{d}{d x} \ln{\left(3 x + 1 \right)} + 2 \ln{\left(3 x + 1 \right)} - 2 \]algebraFactor out the common derivative term.✓ Proved
- \[ = \frac{\left(2 x + \frac{2}{3}\right) \frac{d}{d x} \left(3 x + 1\right)}{3 x + 1} + 2 \ln{\left(3 x + 1 \right)} - 2 \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = \frac{3 \left(2 x + \frac{2}{3}\right)}{3 x + 1} + 2 \ln{\left(3 x + 1 \right)} - 2 \]derivative algebraDifferentiate the inner function 3x + 1. Simplify the product of the fraction and 3.✓ Proved
- \[ = 2 \ln{\left(3 x + 1 \right)} - 2 + \frac{6 x + 2}{3 x + 1} \]algebraDistribute 3 into the numerator.✓ Proved
- \[ = 2 \ln{\left(3 x + 1 \right)} \]algebra algebra algebra simplifyCombine the constant -2 with the fraction using a common denominator. Distribute -2 in the numerator. Simplify the numerator. Final simplification.✓ Proved
Answer \( 2 \ln{\left(3 x + 1 \right)} \)
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 |
| 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 undefined where 3*x + 1 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 1 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*x + 1 = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 15 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 16 | ✓ 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 3 incorrectly labels a distributive expansion as a product rule. The operation (2*x+2/3)*log(3*x+1) → 2*x*log(3*x+1)+(2/3)*log(3*x+1) is an algebraic distribution, not a product rule application. The label should be "algebra" instead of "product".qwen3.6:27b-mlx: fail (error) — Step 3 incorrectly applies the product rule to split a sum. The expression (2*x + 2/3)*log(3*x + 1) is a product of a sum and a log, not a sum of products. The step should have used 'algebra' to distribute the log term into (2*x*log(...) + (2/3)*log(...)) before applying the sum rule, or applied the product rule to the whole term in one go. As written, it claims to apply the product rule to generate a sum, which is mathematically invalid (the product rule yields a sum of products, not a sum of separate derivative terms directly from a single product term without distribution first).
Every verdict on record (4)
gpt-oss:20b: fail (style) 2026-10-10 — Step 3 incorrectly labels a distributive expansion as a product rule. The operation (2*x+2/3)*log(3*x+1) → 2*x*log(3*x+1)+(2/3)*log(3*x+1) is an algebraic distribution, not a product rule application. The label should be "algebra" instead of "product".qwen3.6:27b-mlx: fail (error) 2026-10-10 — Step 3 incorrectly applies the product rule to split a sum. The expression (2*x + 2/3)*log(3*x + 1) is a product of a sum and a log, not a sum of products. The step should have used 'algebra' to distribute the log term into (2*x*log(...) + (2/3)*log(...)) before applying the sum rule, or applied the product rule to the whole term in one go. As written, it claims to apply the product rule to generate a sum, which is mathematically invalid (the product rule yields a sum of products, not a sum of separate derivative terms directly from a single product term without distribution first).gpt-oss:20b: fail (error) 2026-10-10 — Step 3 incorrectly rewrites the derivative of the product (2*x + 2/3)*log(3*x+1) as the sum of derivatives of 2*x*log(3*x+1) and (2/3)*log(3*x+1). This applies two rules at once (splitting the product into a sum before differentiation) and is not a valid application of the product rule. The correct approach is to apply the product rule directly to (2*x + 2/3) and log(3*x+1).qwen3.6:27b-mlx: fail (error) 2026-10-10 — Step 3 incorrectly applies the product rule to split a sum. The term (2*x + 2/3)*log(3*x + 1) is a product of a sum and a log, but the step splits it into Derivative(2*x*log(...)) + Derivative((2/3)*log(...)), which is an application of linearity (sum rule) inside the derivative, not the product rule. The product rule was correctly applied in Step 5 to the term 2*x*log(3*x + 1). Step 3 should have used the 'sum' rule to split the derivative of the product into the sum of derivatives of its parts, or simply distributed the log first. As written, it mislabels a linearity step as a product rule step, and technically applies two operations (distribution/linearity) under the guise of one rule.
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.