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Derivative of \( \displaystyle 2 x \ln{\left(x + 2 \right)} - 2 x + 4 \ln{\left(x + 2 \right)} \)

Problem 2.135 · hard Beautiful Video

Differentiate \( \displaystyle f(x) = 2 x \ln{\left(x + 2 \right)} - 2 x + 4 \ln{\left(x + 2 \right)} \).
  1. \[ \frac{d}{d x} \left(2 x \ln{\left(x + 2 \right)} - 2 x + 4 \ln{\left(x + 2 \right)}\right) \]
    sumStart with the derivative of the entire expression.✓ Proved
  2. \[ = - \frac{d}{d x} 2 x + \frac{d}{d x} 2 x \ln{\left(x + 2 \right)} + \frac{d}{d x} 4 \ln{\left(x + 2 \right)} \]
    sumApply the sum rule to separate the terms.✓ Proved
  3. \[ = \frac{d}{d x} 2 x \ln{\left(x + 2 \right)} + \frac{d}{d x} 4 \ln{\left(x + 2 \right)} - 2 \]
    constantThe derivative of 2*x is 2.✓ Proved
  4. \[ = 2 x \frac{d}{d x} \ln{\left(x + 2 \right)} + \ln{\left(x + 2 \right)} \frac{d}{d x} 2 x + \frac{d}{d x} 4 \ln{\left(x + 2 \right)} - 2 \]
    productApply the product rule to the first term.✓ Proved
  5. \[ = 2 x \frac{d}{d x} \ln{\left(x + 2 \right)} + 2 \ln{\left(x + 2 \right)} + \frac{d}{d x} 4 \ln{\left(x + 2 \right)} - 2 \]
    derivativeThe derivative of 2*x is 2.✓ Proved
  6. \[ = \frac{2 x}{x + 2} + 2 \ln{\left(x + 2 \right)} + 4 \frac{d}{d x} \ln{\left(x + 2 \right)} - 2 \]
    logarithmicThe derivative of log(x + 2) is 1/(x + 2).✓ Proved
  7. \[ = \frac{2 x}{x + 2} + 2 \ln{\left(x + 2 \right)} - 2 + \frac{4}{x + 2} \]
    derivative algebraThe derivative of 4*log(x + 2) is 4/(x + 2). Distribute the constant in the last term.✓ Proved
  8. \[ = 2 \ln{\left(x + 2 \right)} - 2 + \frac{2 x + 4}{x + 2} \]
    algebra algebraCombine the fractions with the same denominator. Factor the numerator of the fraction.✓ Proved
  9. \[ = 2 \ln{\left(x + 2 \right)} \]
    simplify simplifyCancel the common factor (x + 2). Simplify the remaining constant terms.✓ Proved
Answer \( 2 \log{\left(x + 2 \right)} \)
Top: the graph of f = 2x·log(x + 2) - 2x + 4log(x + 2). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 2x/(x + 2) + 2log(x + 2) - 2 + 4/(x + 2).

✓ 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
LineStatusChecked byDetail
1✓ Provedsympy 1.14.0line 1 is the problem as stated
2✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
3✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
4✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
5✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
6✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
undefined where x + 2 = 0
7✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
undefined where x + 2 = 0
8✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
undefined where x + 2 = 0
9✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
undefined where x + 2 = 0
10✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
undefined where x + 2 = 0
11✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
undefined where x + 2 = 0
12✓ Provedsympy 1.14.0simplify(a - b) reduced to 0
log is undefined for non-positive arguments
answer✓ Provedsympy 1.14.0final line against the stated answer: simplify(a - b) reduced to 0
log is undefined for non-positive arguments
answer, a second way✓ Provedsympy 1.14.0SymPy differentiated f directly and got the stated answer

Reviewers

  • gpt-oss:20b: pass
  • deepseek-r1:70b: pass
  • qwen3.6:27b-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.

Senior review claude-sonnet-5-5, 2026-10-04: fail — The math is correct and every line was proved. Step 6, however, does two independent things under a single 'logarithmic' label: it evaluates d/dx log(x+2) in the middle term and pulls the constant 4 out of the last derivative (a constant-multiple step), while the note mentions only the first. Line 7 similarly combines evaluating the last derivative with tidying 2*x*(1/(x+2)) to 2*x/(x+2). These should be split into separate steps.

  • qwen3.6:27b-mlx: dismiss — The coefficient 2*x is kept in line 6 (2*x*(1/(x+2))), and SymPy proved the line equal to line 5, so the claimed mathematical error is false; the real defect in this step is a different one (two changes at once).
  • agent:pipeline: uphold — Beautiful nomination: A messy combination of x*log(x+2), a linear term and a log term collapses, through cancellation of the rational parts, to just 2*log(x+2), because 2x/(x+2) + 4/(x+2) = 2 exactly cancels the -2.
Every verdict on record (16)
  • claude-sonnet-5-5: fail (style) 2026-10-04 — The math is correct and every line was proved. Step 6, however, does two independent things under a single 'logarithmic' label: it evaluates d/dx log(x+2) in the middle term and pulls the constant 4 out of the last derivative (a constant-multiple step), while the note mentions only the first. Line 7 similarly combines evaluating the last derivative with tidying 2*x*(1/(x+2)) to 2*x/(x+2). These should be split into separate steps.
  • 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 2026-09-20
  • qwen3.6:27b-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 2026-09-20
  • qwen3.6:27b-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 2026-09-19
  • gpt-oss:20b: pass 2026-09-19
  • qwen3.6:27b-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 2026-09-19
  • gpt-oss:20b: pass 2026-09-19
  • qwen3.6:27b-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 2026-09-18
  • gpt-oss:20b: pass 2026-09-18
  • gpt-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.

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