Derivative of \( \displaystyle \frac{\operatorname{asin}{\left(4 x \right)}}{2} \)
Problem 2.541 · medium
Differentiate \( \displaystyle f(x) = \frac{\operatorname{asin}{\left(4 x \right)}}{2} \).
- \[ \frac{d}{d x} \frac{\operatorname{asin}{\left(4 x \right)}}{2} \]constant-multipleStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \operatorname{asin}{\left(4 x \right)}}{2} \]chainApply the chain rule to the inner function 4*x.✓ Proved
- \[ = \frac{\frac{d}{d x} 4 x}{2 \sqrt{1 - 16 x^{2}}} \]derivativeDifferentiate the outer function asin(u).✓ Proved
- \[ = \frac{2}{\sqrt{1 - 16 x^{2}}} \]derivative simplifyDifferentiate the inner function 4*x. Simplify the resulting expression.✓ Proved
Answer \( \frac{2}{\sqrt{1 - 16 x^{2}}} \)
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. The reviewers disagree about how one step is explained; every verdict is in the receipt.
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 - 16*x**2 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - 16*x**2 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - 16*x**2 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where 1 - 16*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: passqwen3.6:27b-mlx: fail (style) — Step 1 is labeled 'constant-multiple' but performs no transformation; 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 1/2, which is a 'constant-multiple' step, not a chain rule application.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 1 is labeled 'constant-multiple' but performs no transformation; 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 1/2, which is a 'constant-multiple' step, not a chain rule application.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: fail (error) 2026-09-21 — Step 2 incorrectly labels a constant‑multiple application as a chain rule. Step 3 applies two rules at once (derivative of asin and the chain rule), violating 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.