Home›Calculus 1›Quotient rule›Problem 2.686

Derivative of \( \displaystyle 2 \operatorname{atan}{\left(x \right)} \)

Problem 2.686 · medium Mental math Video

Differentiate \( \displaystyle f(x) = 2 \operatorname{atan}{\left(x \right)} \).

\[ \frac{d}{d x} 2 \operatorname{atan}{\left(x \right)} \]
Start with the derivative of the function.✓ Proved
\[ = 2 \frac{d}{d x} \operatorname{atan}{\left(x \right)} \]
constant-multiplePull the constant out of the derivative.✓ Proved
\[ = \frac{2}{x^{2} + 1} \]
derivative simplifyApply the derivative rule for arctan(x). Simplify the expression.✓ Proved

Answer \( \frac{2}{x^{2} + 1} \)

Top: the graph of f = 2atan(x). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 2/(x² + 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. 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
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x**2 + 1 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x**2 + 1 = 0
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0 undefined where 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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the constant multiple rule and the derivative of arctan, with appropriate labels for each step.

Senior review claude-sonnet-5-5, 2026-10-11: pass — The derivative of 2*atan(x) is 2/(1+x^2). The steps are constant-multiple, then the arctan derivative (labelled 'derivative', which is acceptable; 'inverse-trig' would also fit), then simplify. Each step changes one thing and SymPy proved every line. The panel's objection has no specifics listed, so there is nothing to uphold.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-11 — The derivative of 2*atan(x) is 2/(1+x^2). The steps are constant-multiple, then the arctan derivative (labelled 'derivative', which is acceptable; 'inverse-trig' would also fit), then simplify. Each step changes one thing and SymPy proved every line. The panel's objection has no specifics listed, so there is nothing to uphold.
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the constant multiple rule and the derivative of arctan, with appropriate labels for each step.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: pass 2026-09-21
gpt-oss:20b: pass 2026-09-21
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the constant multiple rule and the derivative of arctan, with appropriate labels and single-rule steps.
gpt-oss:20b: fail (error) 2026-09-21 — Step 1 has no rule label and does not change anything; it should be labeled as a derivative application (or omitted).

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.