Derivative of \( \displaystyle \frac{\operatorname{asin}{\left(3 x \right)}}{3} \)
Problem 2.892 · medium
Differentiate \( \displaystyle f(x) = \frac{\operatorname{asin}{\left(3 x \right)}}{3} \).
- \[ \frac{d}{d x} \frac{\operatorname{asin}{\left(3 x \right)}}{3} \]constant-multipleApply the constant multiple rule.✓ Proved
- \[ = \frac{\frac{d}{d x} \operatorname{asin}{\left(3 x \right)}}{3} \]derivativeDifferentiate the arcsine function.✓ Proved
- \[ = \frac{\frac{d}{d x} 3 x}{3 \sqrt{1 - 9 x^{2}}} \]chainApply the chain rule.✓ Proved
- \[ = \frac{1}{\sqrt{1 - 9 x^{2}}} \]derivative algebra simplifyDifferentiate the inner function 3*x. Simplify the product in the numerator. Cancel the common factor of 3.✓ Proved
Answer \( \frac{1}{\sqrt{1 - 9 x^{2}}} \)
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 - 9*x**2 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - 9*x**2 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - 9*x**2 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - 9*x**2 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where 1 - 9*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 6 attempts to cancel a factor of 3 that no longer exists; the expression after step 5 is already 1/sqrt(1-(3*x)**2). The cancellation step is unnecessary and incorrectly labeled as "simplify".qwen3.6:27b-mlx: pass — The solution correctly applies the constant multiple rule, chain rule, and derivative rules in separate steps. The algebraic simplifications are also handled in distinct steps as required.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the constant multiple rule, chain rule, and derivative rules in separate steps. The algebraic simplifications are also handled in distinct steps as required.gpt-oss:20b: fail (error) 2026-09-26 — Step 6 attempts to cancel a factor of 3 that no longer exists; the expression after step 5 is already 1/sqrt(1-(3*x)**2). The cancellation step is unnecessary and incorrectly labeled as "simplify".qwen3.6:27b-mlx: pass 2026-09-26gpt-oss:20b: pass 2026-09-26
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.