Derivative of \( \displaystyle \frac{\operatorname{asin}{\left(5 x \right)}}{5} \)
Problem 2.977 · medium
Differentiate \( \displaystyle f(x) = \frac{\operatorname{asin}{\left(5 x \right)}}{5} \).
- \[ \frac{d}{d x} \frac{\operatorname{asin}{\left(5 x \right)}}{5} \]constant-multipleApply the constant multiple rule to pull out 1/5.✓ Proved
- \[ = \frac{\frac{d}{d x} \operatorname{asin}{\left(5 x \right)}}{5} \]chainApply the chain rule to the derivative of asin(5*x).✓ Proved
- \[ = \frac{\frac{d}{d x} 5 x}{5 \sqrt{1 - 25 x^{2}}} \]derivativeDifferentiate the inner function 5*x.✓ Proved
- \[ = \frac{1}{\sqrt{1 - 25 x^{2}}} \]simplify simplifyMultiply the remaining terms. Simplify the expression by canceling the 5s.✓ Proved
Answer \( \frac{1}{\sqrt{1 - 25 x^{2}}} \)
✓ 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 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 - 25*x**2 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - 25*x**2 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - 25*x**2 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where 1 - 25*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 (style) — Step 2 is merely a rewrite of the constant‑multiple form and should be labeled "algebra" (or "rewrite"), not "chain". No rule is applied in that line, so the label is incorrect.qwen3.6:27b-mlx: pass — The solution correctly applies the constant multiple rule, chain rule, and derivative of arcsine. Each step isolates a single transformation, and the labels accurately reflect the rules applied.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the constant multiple rule, chain rule, and derivative of arcsine. Each step isolates a single transformation, and the labels accurately reflect the rules applied.gpt-oss:20b: fail (style) 2026-09-27 — Step 2 is merely a rewrite of the constant‑multiple form and should be labeled "algebra" (or "rewrite"), not "chain". No rule is applied in that line, so the label is incorrect.qwen3.6:27b-mlx: pass 2026-09-27gpt-oss:20b: fail (style) 2026-09-27 — The labels for the constant‑multiple and chain rules are swapped. Step 2 should be labeled "constant‑multiple" (pulling out 1/5), and step 3 should be labeled "chain" (applying the chain rule to the inner 5x).
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.