Derivative of \( \displaystyle \frac{\operatorname{asin}{\left(2 x \right)}}{2} \)
Problem 2.507 · medium
Differentiate \( \displaystyle f(x) = \frac{\operatorname{asin}{\left(2 x \right)}}{2} \).
- \[ \frac{d}{d x} \frac{\operatorname{asin}{\left(2 x \right)}}{2} \]constant-multipleStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \operatorname{asin}{\left(2 x \right)}}{2} \]chainApply the chain rule to the inner function 2*x.✓ Proved
- \[ = \frac{\frac{d}{d x} 2 x}{2 \sqrt{1 - 4 x^{2}}} \]derivativeDifferentiate the arcsine function.✓ Proved
- \[ = \frac{1}{\sqrt{1 - 4 x^{2}}} \]derivative simplifyDifferentiate the linear term 2*x. Simplify the expression by canceling the 2 and expanding the square.✓ Proved
Answer \( \frac{1}{\sqrt{1 - 4 x^{2}}} \)
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. A reviewer's objection to the explanation stands; see the receipt. 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 asin is real only on [-1, 1] |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 asin is real only on [-1, 1] undefined where 1 - 4*x**2 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - 4*x**2 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - 4*x**2 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where 1 - 4*x**2 = 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 2 incorrectly labels a constant‑multiple step as "chain". Step 3 applies both the derivative of arcsin and the chain rule for the inner 2*x in a single line, but only labels "derivative", violating the one‑rule‑per‑step rule.qwen3.6:27b-mlx: fail (style) — Step 1 is labeled 'constant-multiple' but performs no operation; it merely restates the problem, violating the rule that each step must change one thing. Step 2 is labeled 'chain' but only factors out the constant, which is a 'constant-multiple' step.
Senior review claude-sonnet-5-5, 2026-10-09: fail — The math is correct and SymPy proved every line. The labels are shifted by one: line 2 (constant-multiple, labeled 'chain') and its note about the inner function 2*x describe the next step, so a student would learn the wrong rule name for pulling out the constant.
gpt-oss:20b: uphold — Line 2 only pulls the factor 1/2 out of the derivative, which is constant-multiple, so the label 'chain' names a rule that step did not apply. The claim that line 3 does two things is wrong: it is a single chain-rule expansion, so 'derivative' is acceptable there.qwen3.6:27b-mlx: uphold — The step into line 2 is constant-multiple but is labeled 'chain', which is a wrong rule name. The complaint about line 1 is not a defect, since line 1 just states the problem.
Every verdict on record (7)
claude-sonnet-5-5: fail (misleading) 2026-10-09 — The math is correct and SymPy proved every line. The labels are shifted by one: line 2 (constant-multiple, labeled 'chain') and its note about the inner function 2*x describe the next step, so a student would learn the wrong rule name for pulling out the constant.qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 1 is labeled 'constant-multiple' but performs no operation; it merely restates the problem, violating the rule that each step must change one thing. Step 2 is labeled 'chain' but only factors out the constant, which is a 'constant-multiple' step.gpt-oss:20b: fail (error) 2026-09-21 — Step 2 incorrectly labels a constant‑multiple step as "chain". Step 3 applies both the derivative of arcsin and the chain rule for the inner 2*x in a single line, but only labels "derivative", violating the one‑rule‑per‑step rule.qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 1 is labeled 'constant-multiple' but performs no operation; it merely restates the problem. Step 2 is labeled 'chain' but only factors out the constant 1/2, which is an application of the constant-multiple rule, not the chain rule.gpt-oss:20b: fail (style) 2026-09-20 — Step 2 incorrectly labels the application of the constant‑multiple rule as "chain". Step 3 applies both the derivative of arcsin and the chain rule in one line, which violates the one‑rule‑per‑step rule and should be split into two separate steps with appropriate labels.qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 1 is labeled 'constant-multiple' but performs no operation; it merely restates the problem, violating the 'one change per step' rule. Step 2 is labeled 'chain' but only factors out the constant 1/2, which is a 'constant-multiple' step, not the chain rule.gpt-oss:20b: fail (error) 2026-09-20 — Step 2 incorrectly labels a constant‑multiple step as "chain". Step 3 applies two rules at once: the derivative of arcsin and the chain rule for the inner function. These multi‑rule steps violate the one‑rule‑per‑step rule.
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.