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Derivative of \( \displaystyle 5 \operatorname{atan}{\left(x + 1 \right)} \)

Problem 2.572 · medium Video

Differentiate \( \displaystyle f(x) = 5 \operatorname{atan}{\left(x + 1 \right)} \).

\[ \frac{d}{d x} 5 \operatorname{atan}{\left(x + 1 \right)} \]
Start with the derivative of the function.✓ Proved
\[ = 5 \frac{d}{d x} \operatorname{atan}{\left(x + 1 \right)} \]
constantPull the constant out of the derivative.✓ Proved
\[ = \frac{5 \frac{d}{d x} \left(x + 1\right)}{\left(x + 1\right)^{2} + 1} \]
inverse-trigApply the derivative rule for arctan.✓ Proved
\[ = \frac{5}{\left(x + 1\right)^{2} + 1} \]
derivative simplifyDifferentiate the inner function x + 1. Simplify the expression.✓ Proved

Answer \( \frac{5}{\left(x + 1\right)^{2} + 1} \)

Top: the graph of f = 5atan(x + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 5/((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
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: fail (error) — Step 3 applies both the inverse‑trig rule for atan and the chain rule for the inner function x+1 in a single line, violating the one‑rule‑per‑step rule. The correct approach would separate the chain rule application into a distinct step.
qwen3.6:27b-mlx: pass

Senior review claude-sonnet-5-5, 2026-10-10: pass — Every line is SymPy-verified and each step makes one change with an acceptable label. Step 3's inverse-trig label is fine, since the chain-rule factor is only written, not evaluated, and step 4 evaluates it.

gpt-oss:20b: dismiss — Step 3 applies the arctan derivative rule to the outer function and leaves Derivative(x + 1, x) unevaluated, so it is one change; the contract treats the chain-rule structure as part of the same step, and the inner derivative is evaluated separately in step 4.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-10 — Every line is SymPy-verified and each step makes one change with an acceptable label. Step 3's inverse-trig label is fine, since the chain-rule factor is only written, not evaluated, and step 4 evaluates it.
qwen3.6:27b-mlx: pass 2026-09-26
gpt-oss:20b: fail (error) 2026-09-26 — Step 3 applies both the inverse‑trig rule for atan and the chain rule for the inner function x+1 in a single line, violating the one‑rule‑per‑step rule. The correct approach would separate the chain rule application into a distinct step.
qwen3.6:27b-mlx: pass 2026-09-21
gpt-oss:20b: pass 2026-09-21
qwen3.6:27b-mlx: pass 2026-09-21
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.