Derivative of \( \displaystyle x \left(\ln{\left(2 x \right)} - 1\right) \)
Problem 2.72 · hard Video
Differentiate \( \displaystyle f(x) = x \left(\ln{\left(2 x \right)} - 1\right) \).
- \[ \frac{d}{d x} x \left(\ln{\left(2 x \right)} - 1\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = x \frac{d}{d x} \left(\ln{\left(2 x \right)} - 1\right) + \left(\ln{\left(2 x \right)} - 1\right) \frac{d}{d x} x \]productApply the product rule.✓ Proved
- \[ = x \frac{d}{d x} \left(\ln{\left(2 x \right)} - 1\right) + \ln{\left(2 x \right)} - 1 \]derivativeDifferentiate the first part of the product.✓ Proved
- \[ = x \left(- \frac{d}{d x} 1 + \frac{d}{d x} \ln{\left(2 x \right)}\right) + \ln{\left(2 x \right)} - 1 \]sumDistribute the derivative over the subtraction.✓ Proved
- \[ = x \frac{d}{d x} \ln{\left(2 x \right)} + \ln{\left(2 x \right)} - 1 \]constantThe derivative of a constant is zero.✓ Proved
- \[ = \ln{\left(2 x \right)} \]chain simplify algebra simplifyApply the chain rule to the logarithm. Simplify the derivative of the log term. Simplify the product of x and 1/x. Combine the constant terms.✓ Proved
Answer \( \log{\left(2 x \right)} \)
✓ 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 |
| 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 |
| 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 (error) — Step 6 applies two rules at once: it uses the chain rule to compute the derivative of log(2*x) and then immediately simplifies 1/(2*x*(1/2)) to 1/x. According to the contract, each step must change only one thing, so this step should be split into a chain step followed by a simplify step.deepseek-r1:70b: passqwen3.6:27b-mlx: fail (error) — Step 6 is mathematically incorrect. The derivative of log(2x) is 1/x, but the step computes 1/(2x * (1/2)) = 1/x via an incorrect chain rule application (the derivative of the inner function 2x is 2, not 1/2). While the final result is correct, the intermediate step contains a factual error in the differentiation process.
Senior review claude-sonnet-5-5, 2026-10-03: pass — The derivation is correct and every line was proved equal to the previous one. Step 6's denominator 2x*(1/2) is an awkward way to write the chain-rule result, which would more naturally be (1/(2x))*2, but it is not false and the label is acceptable.
gpt-oss:20b: dismiss — Step 6 only applies the chain rule, d/dx log(u) = 1/u * u', with u' = 2 evaluated, and the product 2x*(1/2) is left unsimplified; the actual simplification to 1/x is the separate step 7.qwen3.6:27b-mlx: dismiss — The form 1/(2x*(1/2)) is a legitimate if odd chain-rule output (it equals 1/x, and SymPy proved the line), and the claim that it is wrong is unfounded; a factor of 1/2 in the denominator is not the derivative of 2x, but the expression is still mathematically equal to the correct derivative 1/x.
Every verdict on record (13)
claude-sonnet-5-5: pass 2026-10-03 — The derivation is correct and every line was proved equal to the previous one. Step 6's denominator 2x*(1/2) is an awkward way to write the chain-rule result, which would more naturally be (1/(2x))*2, but it is not false and the label is acceptable.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 6 is mathematically incorrect. The derivative of log(2x) is 1/x, but the step computes 1/(2x * (1/2)) = 1/x via an incorrect chain rule application (the derivative of the inner function 2x is 2, not 1/2). While the final result is correct, the intermediate step contains a factual error in the differentiation process.gpt-oss:20b: fail (error) 2026-09-20 — Step 6 applies two rules at once: it uses the chain rule to compute the derivative of log(2*x) and then immediately simplifies 1/(2*x*(1/2)) to 1/x. According to the contract, each step must change only one thing, so this step should be split into a chain step followed by a simplify step.qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the product rule, linearity of differentiation, and the chain rule for the logarithmic term. Each step isolates a single operation and uses appropriate labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies the product rule, linearity of differentiation, and the chain rule for the logarithmic term. Each step isolates a single transformation, and the labels accurately reflect the operations performed.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19gpt-oss:20b: fail 2026-09-17 — Step 6 misapplies the chain rule: it writes 1/(2*x*(1/2)) instead of the correct 1/(2*x)*2, even though the final value happens to be 1/x. This could mislead a student about how the derivative of 2*x is handled.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.