Derivative of \( \displaystyle \ln{\left(\operatorname{atan}{\left(2 x \right)} \right)} \)
Problem 2.465 · hard
Differentiate \( \displaystyle f(x) = \ln{\left(\operatorname{atan}{\left(2 x \right)} \right)} \).
- \[ \frac{d}{d x} \ln{\left(\operatorname{atan}{\left(2 x \right)} \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \operatorname{atan}{\left(2 x \right)}}{\operatorname{atan}{\left(2 x \right)}} \]chainApply the chain rule for the logarithm.✓ Proved
- \[ = \frac{\frac{d}{d x} 2 x}{\left(4 x^{2} + 1\right) \operatorname{atan}{\left(2 x \right)}} \]chainApply the chain rule for the arctangent.✓ Proved
- \[ = \frac{2}{\left(4 x^{2} + 1\right) \operatorname{atan}{\left(2 x \right)}} \]derivative algebraDifferentiate the innermost function 2*x. Simplify the expression by multiplying the terms.✓ Proved
Answer \( \frac{2}{\left(4 x^{2} + 1\right) \operatorname{atan}{\left(2 x \right)}} \)
Mind the domain. The answer is also defined on (-oo, 0), 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) = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where atan(2*x) = 0 undefined where 4*x**2 + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 + 1 = 0 undefined where atan(2*x) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 + 1 = 0 undefined where atan(2*x) = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where 4*x**2 + 1 = 0 undefined where atan(2*x) = 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
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-21gpt-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.