Home›Calculus 1›Derivatives of inverse trig functions›Problem 2.617

Derivative of \( \displaystyle \operatorname{atan}^{2}{\left(x + 1 \right)} \)

Problem 2.617 · medium Video

Differentiate \( \displaystyle f(x) = \operatorname{atan}^{2}{\left(x + 1 \right)} \).

\[ \frac{d}{d x} \operatorname{atan}^{2}{\left(x + 1 \right)} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = 2 \operatorname{atan}{\left(x + 1 \right)} \frac{d}{d x} \operatorname{atan}{\left(x + 1 \right)} \]
powerApply the power rule.✓ Proved
\[ = \frac{2 \operatorname{atan}{\left(x + 1 \right)}}{\left(x + 1\right)^{2} + 1} \]
derivative algebraDifferentiate the inner function using the derivative of arctan. Simplify the expression.✓ Proved
\[ = \frac{2 \operatorname{atan}{\left(x + 1 \right)}}{x^{2} + 2 x + 2} \]
algebra simplifyExpand the denominator. Combine like terms.✓ Proved

Answer \( \frac{2 \operatorname{atan}{\left(x + 1 \right)}}{\left(x + 1\right)^{2} + 1} \)

Top: the graph of f = atan(x + 1)². A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 2atan(x + 1)/((x + 1)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. 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 + 1)**2 + 1 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where (x + 1)**2 + 1 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where (x + 1)2 + 1 = 0 undefined where x2 + 2*x + 2 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x*2 + 2x + 2 = 0
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0 undefined where (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: pass
qwen3.6:27b-mlx: pass

Senior review claude-sonnet-5-5, 2026-10-11: pass — The panel listed no specific objection to rule on. The solution is correct: the power rule gives 2*atan(x+1)*d/dx atan(x+1), the arctan derivative gives 1/(1+(x+1)^2), and the remaining steps are algebra. Every rule label fits its step, and the final line equals the stated answer.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-11 — The panel listed no specific objection to rule on. The solution is correct: the power rule gives 2*atan(x+1)*d/dx atan(x+1), the arctan derivative gives 1/(1+(x+1)^2), and the remaining steps are algebra. Every rule label fits its step, and the final line equals the stated answer.
qwen3.6:27b-mlx: pass 2026-09-26
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 2 applies the power rule but fails to apply the chain rule for the inner function (x+1), incorrectly leaving a Derivative term instead of computing the derivative of the inner function. This violates the 'one rule per step' constraint by conflating the power rule with an incomplete chain rule application.
gpt-oss:20b: pass 2026-09-21
qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 2 applies both the power rule and the chain rule simultaneously, violating the one-rule-per-step constraint. It should be split into a power rule step leaving the derivative operator, followed by a chain rule step.
gpt-oss:20b: pass 2026-09-21

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.