Derivative of \( \displaystyle 3 x \ln{\left(2 x + 2 \right)} - 3 x + 3 \ln{\left(x + 1 \right)} \)
Problem 2.2520 · hard Video
Differentiate \( \displaystyle f(x) = 3 x \ln{\left(2 x + 2 \right)} - 3 x + 3 \ln{\left(x + 1 \right)} \).
- \[ \frac{d}{d x} \left(3 x \ln{\left(2 x + 2 \right)} - 3 x + 3 \ln{\left(x + 1 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{d}{d x} 3 x + \frac{d}{d x} 3 x \ln{\left(2 x + 2 \right)} + \frac{d}{d x} 3 \ln{\left(x + 1 \right)} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} 3 x \ln{\left(2 x + 2 \right)} + \frac{d}{d x} 3 \ln{\left(x + 1 \right)} - 3 \]constantDifferentiate the constant term.✓ Proved
- \[ = 3 \frac{d}{d x} x \ln{\left(2 x + 2 \right)} + 3 \frac{d}{d x} \ln{\left(x + 1 \right)} - 3 \]constant-multipleFactor out the constant 3.✓ Proved
- \[ = 3 x \frac{d}{d x} \ln{\left(2 x + 2 \right)} + 3 \ln{\left(2 x + 2 \right)} \frac{d}{d x} x + 3 \frac{d}{d x} \ln{\left(x + 1 \right)} - 3 \]productApply the product rule to the first term.✓ Proved
- \[ = \frac{3 x \frac{d}{d x} \left(2 x + 2\right)}{2 x + 2} + 3 \ln{\left(2 x + 2 \right)} + 3 \frac{d}{d x} \ln{\left(x + 1 \right)} - 3 \]chainApply the chain rule to the log term.✓ Proved
- \[ = \frac{6 x}{2 x + 2} + 3 \ln{\left(2 x + 2 \right)} + 3 \frac{d}{d x} \ln{\left(x + 1 \right)} - 3 \]derivative algebraDifferentiate the inner function 2x + 2. Distribute the 3 into the parentheses.✓ Proved
- \[ = \frac{3 x}{x + 1} + 3 \ln{\left(2 x + 2 \right)} + 3 \frac{d}{d x} \ln{\left(x + 1 \right)} - 3 \]algebraSimplify the fraction 2/(2x + 2) to 1/(x + 1).✓ Proved
- \[ = \frac{3 x}{x + 1} + 3 \ln{\left(2 x + 2 \right)} - 3 + \frac{3}{x + 1} \]chain algebraDifferentiate the log term. Simplify the product.✓ Proved
- \[ = 3 \ln{\left(2 x + 2 \right)} - 3 + \frac{3 x + 3}{x + 1} \]algebra algebraCombine the fractions with common denominator x + 1. Factor out the 3 from the numerator.✓ Proved
- \[ = 3 \ln{\left(2 x + 2 \right)} \]algebra simplifySimplify the fraction (x + 1)/(x + 1). Combine the constant terms.✓ Proved
Answer \( 3 \ln{\left(2 x + 2 \right)} \)
✓ 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. How we verify
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 |
| 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 x + 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments 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: passqwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and algebraic simplifications in a step-by-step manner. Each step changes only one aspect of the expression and uses an appropriate label from the fixed vocabulary.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-29 — The solution correctly applies differentiation rules and algebraic simplifications in a step-by-step manner. Each step changes only one aspect of the expression and uses an appropriate label from the fixed vocabulary.gpt-oss:20b: pass 2026-10-29qwen3.6:27b-mlx: pass 2026-10-28 — 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: fail (style) 2026-10-28 — Step 3 incorrectly labels the operation as a "constant" rule. The term 3*x is not a constant; its derivative is obtained by the product rule (constant‑multiple rule) applied to 3 and x, yielding 3. The label should be "constant‑multiple" or "product" rather than "constant".qwen3.6:27b-mlx: pass 2026-10-28 — The solution correctly applies differentiation rules one at a time and simplifies the result accurately. All labels correspond to the actions taken in each step.gpt-oss:20b: pass 2026-10-28
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 2026-10-29 with SymPy 1.14.0.