Home›Calculus 1›Derivatives of inverse trig functions›Problem 2.827

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

Problem 2.827 · medium Video

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

\[ \frac{d}{d x} \operatorname{atan}^{2}{\left(x + 2 \right)} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = 2 \operatorname{atan}{\left(x + 2 \right)} \frac{d}{d x} \operatorname{atan}{\left(x + 2 \right)} \]
powerApply the power rule.✓ Proved
\[ = \frac{2 \operatorname{atan}{\left(x + 2 \right)}}{\left(x + 2\right)^{2} + 1} \]
derivative algebraDifferentiate the inner function using the derivative of arctan. Simplify the expression.✓ Proved
\[ = \frac{2 \operatorname{atan}{\left(x + 2 \right)}}{x^{2} + 4 x + 5} \]
algebra simplifyExpand the denominator. Combine like terms.✓ Proved

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

Top: the graph of f = atan(x + 2)². A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 2atan(x + 2)/((x + 2)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 + 2)**2 + 1 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where (x + 2)**2 + 1 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where (x + 2)2 + 1 = 0 undefined where x2 + 4*x + 5 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x*2 + 4x + 5 = 0
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0 undefined where (x + 2)**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: fail (error) — Step 2 applies the power rule but fails to apply the chain rule to the inner function `atan(x+2)`, leaving a `Derivative` operator that is not resolved by the power rule alone. The transition from Step 1 to Step 2 requires both the power rule and the chain rule, violating the one-rule-per-step constraint.

Senior review claude-sonnet-5-5, 2026-10-13: pass — All lines are equivalent (SymPy proved each), the rule labels are legitimate, and each step changes one thing. Evaluating atan' in step 3 and expanding the denominator in step 5 are separate single changes.

qwen3.6:27b-mlx: dismiss — Step 2 applies the power rule to the outer square, leaving Derivative(atan(x+2), x) to be evaluated in the next step; the chain factor is just that remaining derivative, so the step changes one thing and the label is legitimate.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-13 — All lines are equivalent (SymPy proved each), the rule labels are legitimate, and each step changes one thing. Evaluating atan' in step 3 and expanding the denominator in step 5 are separate single changes.
qwen3.6:27b-mlx: fail (error) 2026-09-27 — Step 2 applies the power rule but fails to apply the chain rule to the inner function `atan(x+2)`, leaving a `Derivative` operator that is not resolved by the power rule alone. The transition from Step 1 to Step 2 requires both the power rule and the chain rule, violating the one-rule-per-step constraint.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: pass 2026-09-26
gpt-oss:20b: pass 2026-09-26

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 2026-09-27 with SymPy 1.14.0.