Derivative of \( \displaystyle - x \ln{\left(x \right)} + x \)
Problem 2.2106 · hard
Differentiate \( \displaystyle f(x) = - x \ln{\left(x \right)} + x \).
- \[ \frac{d}{d x} \left(- x \ln{\left(x \right)} + x\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} x + \frac{d}{d x} \left(- x \ln{\left(x \right)}\right) \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \left(- x \ln{\left(x \right)}\right) + 1 \]constantThe derivative of x is 1.✓ Proved
- \[ = - x \frac{d}{d x} \ln{\left(x \right)} - \ln{\left(x \right)} + 1 \]productApply the product rule to the first term.✓ Proved
- \[ = - \ln{\left(x \right)} \]derivative algebra simplifyThe derivative of log(x) is 1/x. Simplify the term x*(1/x). Combine the constant terms.✓ Proved
Answer \( - \ln{\left(x \right)} \)
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 |
| 7 | ✓ 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: fail (error) — Step 3 is labeled 'constant', but it applies the power rule (or derivative of x) to compute the derivative of x as 1. The 'constant' rule applies to derivatives of constant values, not to the variable x.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-12 — Step 3 is labeled 'constant', but it applies the power rule (or derivative of x) to compute the derivative of x as 1. The 'constant' rule applies to derivatives of constant values, not to the variable x.gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: pass 2026-10-12gpt-oss:20b: pass 2026-10-12
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 2026-10-12 with SymPy 1.14.0.