Derivative of \( \displaystyle \operatorname{atan}{\left(\sin{\left(x \right)} \right)} \)
Problem 2.583 · hard
Differentiate \( \displaystyle f(x) = \operatorname{atan}{\left(\sin{\left(x \right)} \right)} \).
- \[ \frac{d}{d x} \operatorname{atan}{\left(\sin{\left(x \right)} \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \sin{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} \]chainApply the chain rule for the arctangent function.✓ Proved
- \[ = \frac{\cos{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} \]productDifferentiate the sine function.✓ Proved
Answer \( \frac{\cos{\left(x \right)}}{\sin^{2}{\left(x \right)} + 1} \)
✓ 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 sin(x)**2 + 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(x)**2 + 1 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where sin(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: fail (style) — Step 3 is labeled "product" but only the derivative of sin(x) was applied; no product rule was used. The correct label would be "derivative" (or "simplify" after multiplication).qwen3.6:27b-mlx: fail (style) — Step 3 is labeled 'product', but the step actually computes the derivative of sin(x) (a 'derivative' step) and simplifies the expression. The product rule is not applied here; the chain rule application in step 2 already set up the multiplication, and step 3 just resolves the inner derivative.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 3 is labeled 'product', but the step actually computes the derivative of sin(x) (a 'derivative' step) and simplifies the expression. The product rule is not applied here; the chain rule application in step 2 already set up the multiplication, and step 3 just resolves the inner derivative.gpt-oss:20b: fail (style) 2026-09-21 — Step 3 is labeled "product" but only the derivative of sin(x) was applied; no product rule was used. The correct label would be "derivative" (or "simplify" after multiplication).qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 3 applies the derivative of sine but is labeled 'product'. The step should be labeled 'derivative' or 'chain' (if viewed as part of the chain rule application), but 'product' is incorrect as no product rule is applied.gpt-oss:20b: fail (style) 2026-09-21 — Step 3 incorrectly labels the application of the derivative of sin(x) as "product". The correct rule name is "derivative"; no product rule is used here.
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.