Derivative of \( \displaystyle \ln{\left(\frac{e^{- 2 x}}{2 x} \right)} \)
Problem 2.1397 · hard Beautiful
Differentiate \( \displaystyle f(x) = \ln{\left(\frac{e^{- 2 x}}{2 x} \right)} \).
- \[ \frac{d}{d x} \ln{\left(\frac{e^{- 2 x}}{2 x} \right)} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- 2 x - \ln{\left(2 x \right)}\right) \]algebra simplifyUse the property log(a/b) = log(a) - log(b). Simplify log(exp(-2*x)) to -2*x.✓ Proved
- \[ = \frac{d}{d x} \left(- 2 x\right) - \frac{d}{d x} \ln{\left(2 x \right)} \]sumSplit the derivative into two parts.✓ Proved
- \[ = - \frac{d}{d x} \ln{\left(2 x \right)} - 2 \]constantThe derivative of -2*x is -2.✓ Proved
- \[ = -2 - \frac{\frac{d}{d x} 2 x}{2 x} \]chainApply the chain rule to log(2*x).✓ Proved
- \[ = -2 - \frac{1}{x} \]derivative algebraThe derivative of 2*x is 2. Simplify the expression.✓ Proved
- \[ = \frac{- 2 x - 1}{x} \]simplifyCombine into a single fraction.✓ Proved
Answer \( -2 - \frac{1}{x} \)
Mind the domain. The answer is also defined on (-oo, 0), where f(x) is not. Substituting there gives a number that is not a slope of f.
✓ 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 undefined where x = 0 |
| 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 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where x = 0 |
| 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 logarithmic properties, simplification, and differentiation rules. Each step isolates a single transformation, and the labels accurately reflect the operations performed.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies logarithmic properties, simplification, and differentiation rules. Each step isolates a single transformation, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: fail (style) 2026-10-03 — Step 5 is labeled "constant" but the operation performed is the derivative of -2*x, which should be labeled "derivative". All other steps are correctly labeled and apply only one rule each.
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 2026-10-03 with SymPy 1.14.0.