Derivative of \( \displaystyle \ln{\left(\operatorname{atan}{\left(2 x - 1 \right)} \right)} \)
Problem 2.484 · hard
Differentiate \( \displaystyle f(x) = \ln{\left(\operatorname{atan}{\left(2 x - 1 \right)} \right)} \).
- \[ \frac{d}{d x} \ln{\left(\operatorname{atan}{\left(2 x - 1 \right)} \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \operatorname{atan}{\left(2 x - 1 \right)}}{\operatorname{atan}{\left(2 x - 1 \right)}} \]chainApply the chain rule for the logarithm.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(2 x - 1\right)}{\left(\left(2 x - 1\right)^{2} + 1\right) \operatorname{atan}{\left(2 x - 1 \right)}} \]chainApply the chain rule for the arctangent.✓ Proved
- \[ = \frac{2}{\left(\left(2 x - 1\right)^{2} + 1\right) \operatorname{atan}{\left(2 x - 1 \right)}} \]derivative algebraDifferentiate the linear term inside. Combine the terms into a single fraction.✓ Proved
- \[ = \frac{2}{\left(4 x^{2} - 4 x + 2\right) \operatorname{atan}{\left(2 x - 1 \right)}} \]algebra simplifyExpand the squared term. Simplify the expression in the denominator.✓ Proved
Answer \( \frac{2}{\left(\left(2 x - 1\right)^{2} + 1\right) \operatorname{atan}{\left(2 x - 1 \right)}} \)
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 undefined where atan(2*x - 1) = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where atan(2*x - 1) = 0 undefined where (2*x - 1)**2 + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where atan(2*x - 1) = 0 undefined where (2*x - 1)**2 + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where atan(2*x - 1) = 0 undefined where (2*x - 1)**2 + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where atan(2*x - 1) = 0 undefined where (2*x - 1)**2 + 1 = 0 undefined where 4*x**2 - 4*x + 2 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 - 4*x + 2 = 0 undefined where atan(2*x - 1) = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where atan(2*x - 1) = 0 undefined where (2*x - 1)**2 + 1 = 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 the chain rule in sequence for the logarithm, arctangent, and linear inner function. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the chain rule in sequence for the logarithm, arctangent, and linear inner function. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.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.