Derivative of \( \displaystyle 2 x \ln{\left(2 x + 2 \right)} - 2 x + 2 \ln{\left(x + 1 \right)} \)
Problem 2.136 · hard
Differentiate \( \displaystyle f(x) = 2 x \ln{\left(2 x + 2 \right)} - 2 x + 2 \ln{\left(x + 1 \right)} \).
- \[ \frac{d}{d x} \left(2 x \ln{\left(2 x + 2 \right)} - 2 x + 2 \ln{\left(x + 1 \right)}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} 2 x + \frac{d}{d x} 2 x \ln{\left(2 x + 2 \right)} + \frac{d}{d x} 2 \ln{\left(x + 1 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} 2 x \ln{\left(2 x + 2 \right)} + \frac{d}{d x} 2 \ln{\left(x + 1 \right)} - 2 \]constantDifferentiate the term -2*x.✓ Proved
- \[ = 2 x \frac{d}{d x} \ln{\left(2 x + 2 \right)} + \ln{\left(2 x + 2 \right)} \frac{d}{d x} 2 x + \frac{d}{d x} 2 \ln{\left(x + 1 \right)} - 2 \]productApply the product rule to the first term.✓ Proved
- \[ = 2 x \frac{d}{d x} \ln{\left(2 x + 2 \right)} + 2 \ln{\left(2 x + 2 \right)} + \frac{d}{d x} 2 \ln{\left(x + 1 \right)} - 2 \]constant-multipleDifferentiate the term 2*x.✓ Proved
- \[ = \frac{2 x \frac{d}{d x} \left(2 x + 2\right)}{2 x + 2} + 2 \ln{\left(2 x + 2 \right)} + \frac{d}{d x} 2 \ln{\left(x + 1 \right)} - 2 \]chainApply the chain rule to log(2*x + 2).✓ Proved
- \[ = \frac{4 x}{2 x + 2} + 2 \ln{\left(2 x + 2 \right)} + \frac{d}{d x} 2 \ln{\left(x + 1 \right)} - 2 \]derivativeDifferentiate the inner function 2*x + 2.✓ Proved
- \[ = \frac{4 x}{2 x + 2} + 2 \ln{\left(2 x + 2 \right)} - 2 + \frac{2 \frac{d}{d x} \left(x + 1\right)}{x + 1} \]constant-multiplePull the constant 2 out of the last derivative.✓ Proved
- \[ = \frac{4 x}{2 x + 2} + 2 \ln{\left(2 x + 2 \right)} - 2 + \frac{2}{x + 1} \]derivative algebraDifferentiate the inner function x + 1. Simplify the derivative of log(x+1).✓ Proved
- \[ = \frac{2 x}{x + 1} + 2 \ln{\left(2 x + 2 \right)} - 2 + \frac{2}{x + 1} \]algebraSimplify the term 2/(2*x + 2) * 2.✓ Proved
- \[ = 2 \ln{\left(2 x + 2 \right)} - 2 + \frac{2 x + 2}{x + 1} \]algebra algebraCombine the fractions with common denominator x + 1. Factor the numerator.✓ Proved
- \[ = 2 \ln{\left(2 x + 2 \right)} \]algebra simplifySimplify the fraction. Final simplification.✓ Proved
Answer \( 2 \log{\left(2 x + 2 \right)} \)
✓ 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 undefined where 2*x + 2 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x + 2 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x + 2 = 0 undefined where x + 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x + 2 = 0 undefined where x + 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x + 2 = 0 undefined where x + 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x + 2 = 0 undefined where x + 1 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x + 1 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x + 1 = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x + 1 = 0 |
| 15 | ✓ 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: passdeepseek-r1:70b: fail (style) — Step 1 is missing a rule label, which is required by the contract.qwen3.6:27b-mlx: pass — 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.
Every verdict on record (15)
qwen3.6:27b-mlx: pass 2026-09-20 — 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.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20 — 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.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The algebraic simplifications are valid and clearly labeled.deepseek-r1:70b: fail (style) 2026-09-19 — Step 1 is missing a rule label, which is required by the contract.gpt-oss:20b: fail (style) 2026-09-19 — Step 5 incorrectly labels the application of the derivative rule as "constant-multiple". The step replaces Derivative(2*x,x) with 2, which is a direct application of the derivative rule, not a constant‑multiple simplification. No other multi‑rule steps are present.qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels accurately describe the operations performed, and the algebraic simplifications are valid.deepseek-r1:70b: fail (misleading) 2026-09-19 — Step 3 incorrectly labels the differentiation of -2x as 'constant' instead of 'constant-multiple', which could mislead a student.gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-18 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels accurately describe the operations performed, and the algebraic simplifications are valid.deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: pass 2026-09-18gpt-oss:20b: fail 2026-09-17 — Step 11’s note incorrectly describes the term being simplified – it states "2/(2*x + 2) * 2" while the actual expression is "2*x*(1/(2*x+2))*2". This misleads the student about what is being simplified.deepseek-r1:70b: pass 2026-09-17
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 2026-09-26 with SymPy 1.14.0.