Derivative of \( \displaystyle x - \ln{\left(\left(2 x\right)^{x} \right)} \)
Problem 2.2475 · hard Video
Differentiate \( \displaystyle f(x) = x - \ln{\left(\left(2 x\right)^{x} \right)} \).
- \[ \frac{d}{d x} \left(x - \ln{\left(\left(2 x\right)^{x} \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} x - \frac{d}{d x} \ln{\left(\left(2 x\right)^{x} \right)} \]sumApply the sum rule.✓ Proved
- \[ = 1 - \frac{d}{d x} \ln{\left(\left(2 x\right)^{x} \right)} \]constantDifferentiate the constant x.✓ Proved
- \[ = 1 - \frac{d}{d x} x \ln{\left(2 x \right)} \]rewriteRewrite the logarithm of a power using the property log(a^b) = b*log(a).✓ Proved
- \[ = - x \frac{d}{d x} \ln{\left(2 x \right)} - \ln{\left(2 x \right)} \frac{d}{d x} x + 1 \]productApply the product rule to x * log(2*x).✓ Proved
- \[ = - x \frac{d}{d x} \ln{\left(2 x \right)} - \ln{\left(2 x \right)} + 1 \]constantDifferentiate x.✓ Proved
- \[ = - \ln{\left(2 x \right)} - \frac{\frac{d}{d x} 2 x}{2} + 1 \]chainApply the chain rule to log(2*x).✓ Proved
- \[ = - \ln{\left(2 x \right)} \]derivative algebra algebra algebraDifferentiate 2*x. Simplify the expression. Simplify the expression. Subtract 1.✓ Proved
Answer \( - \ln{\left(2 x \right)} \)
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. 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 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 11 | ✓ 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 9 incorrectly simplifies (2/(2*x))*x to 2/2, applying two simplifications at once and yielding an intermediate expression that is not algebraically equivalent to the previous line. The correct simplification is (2/(2*x))*x = 1, which would still lead to the same final result but the step as written is invalid.qwen3.6:27b-mlx: fail (error) — Step 3 incorrectly labels the differentiation of x as 'constant'; x is not a constant. Step 6 also incorrectly labels the differentiation of x as 'constant'.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-24 — Step 3 incorrectly labels the differentiation of x as 'constant'; x is not a constant. Step 6 also incorrectly labels the differentiation of x as 'constant'.gpt-oss:20b: fail (error) 2026-10-24 — Step 9 incorrectly simplifies (2/(2*x))*x to 2/2, applying two simplifications at once and yielding an intermediate expression that is not algebraically equivalent to the previous line. The correct simplification is (2/(2*x))*x = 1, which would still lead to the same final result but the step as written is invalid.qwen3.6:27b-mlx: fail (error) 2026-10-23 — Step 3 incorrectly labels the differentiation of x as 'constant'; the derivative of x is 1, not a constant rule application. Step 6 also mislabels the differentiation of x as 'constant'.gpt-oss:20b: fail (error) 2026-10-23 — Step 3 incorrectly labels the differentiation of x as a constant; x is a variable. Step 9 applies an algebraic simplification that drops the factor x, incorrectly reducing (2/(2*x))*x to 2/2. These misapplications violate the rule‑by‑rule granularity requirement.qwen3.6:27b-mlx: pass 2026-10-23 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint and using valid labels from the fixed vocabulary.gpt-oss:20b: fail (style) 2026-10-23 — Steps 3 and 6 incorrectly label the derivative of x as "constant"; the correct label is "derivative". The notes also incorrectly describe x as a constant.
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 2026-10-24 with SymPy 1.14.0.