Derivative of \( \displaystyle x \left(1 - \ln{\left(x \right)}\right) \)
Problem 2.482 · medium
Differentiate \( \displaystyle f(x) = x \left(1 - \ln{\left(x \right)}\right) \).
- \[ \frac{d}{d x} x \left(1 - \ln{\left(x \right)}\right) \]Start with the derivative of the function.✓ Proved
- \[ = x \frac{d}{d x} \left(1 - \ln{\left(x \right)}\right) + \left(1 - \ln{\left(x \right)}\right) \frac{d}{d x} x \]productApply the product rule.✓ Proved
- \[ = x \frac{d}{d x} \left(1 - \ln{\left(x \right)}\right) - \ln{\left(x \right)} + 1 \]derivativeDifferentiate the first part of the product.✓ Proved
- \[ = x \left(\frac{d}{d x} 1 - \frac{d}{d x} \ln{\left(x \right)}\right) - \ln{\left(x \right)} + 1 \]sumDifferentiate the second part of the product.✓ Proved
- \[ = - x \frac{d}{d x} \ln{\left(x \right)} - \ln{\left(x \right)} + 1 \]derivativeDifferentiate the terms inside the parentheses.✓ Proved
- \[ = - \ln{\left(x \right)} \]derivative algebra algebra simplifyDifferentiate the natural logarithm. Distribute x into the parentheses. Simplify the fraction x/x. Combine the constant terms.✓ Proved
Answer \( - \log{\left(x \right)} \)
✓ 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.
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: passqwen3.6:27b-mlx: pass — The solution correctly applies the product rule, sum rule, and derivative rules for constants and logarithms. Each step isolates a single transformation, and the labels accurately reflect the operations performed.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the product rule, sum rule, and derivative rules for constants and logarithms. Each step isolates a single transformation, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20
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.