Derivative of \( \displaystyle 2 x \ln{\left(x + 2 \right)} - 2 x + 4 \ln{\left(x + 2 \right)} \)
Problem 2.135 · hard Beautiful
Differentiate \( \displaystyle f(x) = 2 x \ln{\left(x + 2 \right)} - 2 x + 4 \ln{\left(x + 2 \right)} \).
- \[ \frac{d}{d x} \left(2 x \ln{\left(x + 2 \right)} - 2 x + 4 \ln{\left(x + 2 \right)}\right) \]sumStart with the derivative of the entire expression.✓ Proved
- \[ = - \frac{d}{d x} 2 x + \frac{d}{d x} 2 x \ln{\left(x + 2 \right)} + \frac{d}{d x} 4 \ln{\left(x + 2 \right)} \]sumApply the sum rule to separate the terms.✓ Proved
- \[ = \frac{d}{d x} 2 x \ln{\left(x + 2 \right)} + \frac{d}{d x} 4 \ln{\left(x + 2 \right)} - 2 \]constantThe derivative of 2*x is 2.✓ Proved
- \[ = 2 x \frac{d}{d x} \ln{\left(x + 2 \right)} + \ln{\left(x + 2 \right)} \frac{d}{d x} 2 x + \frac{d}{d x} 4 \ln{\left(x + 2 \right)} - 2 \]productApply the product rule to the first term.✓ Proved
- \[ = 2 x \frac{d}{d x} \ln{\left(x + 2 \right)} + 2 \ln{\left(x + 2 \right)} + \frac{d}{d x} 4 \ln{\left(x + 2 \right)} - 2 \]derivativeThe derivative of 2*x is 2.✓ Proved
- \[ = \frac{2 x}{x + 2} + 2 \ln{\left(x + 2 \right)} + 4 \frac{d}{d x} \ln{\left(x + 2 \right)} - 2 \]logarithmicThe derivative of log(x + 2) is 1/(x + 2).✓ Proved
- \[ = \frac{2 x}{x + 2} + 2 \ln{\left(x + 2 \right)} - 2 + \frac{4}{x + 2} \]derivative algebraThe derivative of 4*log(x + 2) is 4/(x + 2). Distribute the constant in the last term.✓ Proved
- \[ = 2 \ln{\left(x + 2 \right)} - 2 + \frac{2 x + 4}{x + 2} \]algebra algebraCombine the fractions with the same denominator. Factor the numerator of the fraction.✓ Proved
- \[ = 2 \ln{\left(x + 2 \right)} \]simplify simplifyCancel the common factor (x + 2). Simplify the remaining constant terms.✓ Proved
Answer \( 2 \log{\left(x + 2 \right)} \)
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 undefined where x + 2 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x + 2 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x + 2 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x + 2 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x + 2 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x + 2 = 0 |
| 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: passdeepseek-r1:70b: passqwen3.6:27b-mlx: fail (error) — Step 6 incorrectly applies the logarithmic rule to the term 2*x*Derivative(log(x + 2), x) by replacing the derivative with 1/(x+2) while ignoring the coefficient 2*x, effectively treating it as if the coefficient were 1 or applying the rule to the wrong term. The step should have only evaluated the derivative in the last term or used constant-multiple/chain rules properly for the middle term.
Every verdict on record (15)
qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 6 incorrectly applies the logarithmic rule to the term 2*x*Derivative(log(x + 2), x) by replacing the derivative with 1/(x+2) while ignoring the coefficient 2*x, effectively treating it as if the coefficient were 1 or applying the rule to the wrong term. The step should have only evaluated the derivative in the last term or used constant-multiple/chain rules properly for the middle term.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 3 incorrectly labels the differentiation of 2*x as 'constant'; the derivative of 2*x is 2, but 2*x is not a constant, so the rule applied is 'derivative' (or 'constant-multiple' combined with 'derivative', but 'constant' is factually wrong). Step 6 incorrectly labels the step as 'logarithmic' while also applying the constant multiple rule to the last term (changing 4*Derivative(...) to 4*...), violating the one-rule-per-step constraint.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 6 is labeled 'logarithmic' but applies the constant-multiple rule to the last term (pulling out the 4) while simultaneously applying the chain/logarithmic rule to the middle term. This violates the one-rule-per-step constraint. Additionally, Step 7 is labeled 'derivative' but performs algebraic simplification (distributing the 4), which is a mislabeling.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 3 claims to apply the 'constant' rule to differentiate 2*x, but 2*x is not a constant; it should be labeled 'derivative' or 'constant-multiple'. Step 6 incorrectly labels the application of the chain rule (differentiating log(x+2)) as 'logarithmic'; the 'logarithmic' rule typically refers to the base derivative d/dx(log(x))=1/x, whereas handling the inner function (x+2) requires the 'chain' rule.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: fail (error) 2026-09-18 — Step 3 applies the derivative to the term 2*x but fails to apply the constant multiple rule, resulting in the derivative of 2*x being written as 2 instead of 2*1 or just keeping the derivative operator. More critically, Step 6 claims the rule is 'logarithmic' but the step actually applies the constant multiple rule to the last term (pulling out the 4) while simultaneously evaluating the derivative of log(x+2) in the middle term. This violates the 'one rule per step' constraint by mixing constant-multiple extraction with derivative evaluation across different terms.deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: pass 2026-09-18gpt-oss:20b: fail 2026-09-17 — Step 3 incorrectly labels the rule as "constant"; the derivative of 2*x is obtained via the constant‑multiple rule, not a constant‑function rule, which could mislead a student.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.