Derivative of \( \displaystyle \operatorname{atan}{\left(2 x \right)} \)
Problem 2.323 · easy
Differentiate \( \displaystyle f(x) = \operatorname{atan}{\left(2 x \right)} \).
- \[ \frac{d}{d x} \operatorname{atan}{\left(2 x \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} 2 x}{4 x^{2} + 1} \]inverse-trig algebraApply the derivative rule for arctangent. Expand the squared term.✓ Proved
- \[ = \frac{2}{4 x^{2} + 1} \]derivative simplifyDifferentiate the inner function 2*x. Simplify the final expression.✓ Proved
Answer \( \frac{2}{4 x^{2} + 1} \)
✓ 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 undefined where 4*x**2 + 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 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 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 + 1 = 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 |
| 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 chain rule via the inverse-trig derivative formula, handles the algebraic expansion and inner derivative separately, and simplifies the result. Each step changes only one thing and uses valid labels.
Every verdict on record (13)
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the chain rule via the inverse-trig derivative formula, handles the algebraic expansion and inner derivative separately, and simplifies the result. Each step changes only one thing and uses valid labels.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the chain rule via the inverse-trig derivative formula, handles the algebraic expansion and inner derivative in separate steps, and simplifies correctly.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-18 — The solution correctly applies the chain rule via the inverse-trig derivative formula, followed by algebraic simplification and differentiation of the inner linear term. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: pass 2026-09-18
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.