Derivative of \( \displaystyle - \frac{\ln{\left(2 x - 1 \right)}}{2} \)
Problem 2.246 · medium
Differentiate \( \displaystyle f(x) = - \frac{\ln{\left(2 x - 1 \right)}}{2} \).
- \[ \frac{d}{d x} \left(- \frac{\ln{\left(2 x - 1 \right)}}{2}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{\frac{d}{d x} \ln{\left(2 x - 1 \right)}}{2} \]constant-multiplePull out the constant factor -1/2.✓ Proved
- \[ = - \frac{\frac{d}{d x} \left(2 x - 1\right)}{2 \left(2 x - 1\right)} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = - \frac{1}{2 x - 1} \]derivative algebraDifferentiate the inner function 2*x - 1. Simplify the expression by canceling the 2.✓ Proved
Answer \( - \frac{1}{2 x - 1} \)
Mind the domain. The answer is also defined on (-oo, 1/2), 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 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 2*x - 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x - 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x - 1 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where 2*x - 1 = 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: passdeepseek-r1:70b: passqwen3.6:27b-mlx: pass — The solution correctly applies the constant multiple rule, chain rule, and derivative rules in separate steps. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.
Every verdict on record (12)
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the constant multiple rule, chain rule, and derivative rules in separate steps. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies the constant multiple rule, chain rule, and power rule in separate steps. The labels are accurate and the logic is sound.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies the constant multiple rule, chain rule, and derivative rules in separate steps. The final simplification is algebraic and correctly labeled.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19gpt-oss:20b: pass 2026-09-17deepseek-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.