Derivative of \( \displaystyle \operatorname{asin}{\left(4 x + 2 \right)} \)
Problem 2.821 · easy
Differentiate \( \displaystyle f(x) = \operatorname{asin}{\left(4 x + 2 \right)} \).
- \[ \frac{d}{d x} \operatorname{asin}{\left(4 x + 2 \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(4 x + 2\right)}{\sqrt{1 - \left(4 x + 2\right)^{2}}} \]derivativeApply the derivative rule for arcsine.✓ Proved
- \[ = \frac{4}{\sqrt{1 - \left(4 x + 2\right)^{2}}} \]derivative algebraDifferentiate the inner function 4*x + 2. Combine the terms into a single fraction.✓ Proved
- \[ = \frac{4}{\sqrt{- 16 x^{2} - 16 x - 3}} \]algebra simplifyExpand the squared term inside the square root. Simplify the expression under the square root.✓ Proved
Answer \( \frac{4}{\sqrt{1 - \left(4 x + 2\right)^{2}}} \)
✓ 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] undefined where 1 - (4*x + 2)**2 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - (4*x + 2)**2 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - (4*x + 2)**2 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - (4*x + 2)**2 = 0 undefined where -16*x**2 - 16*x - 3 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where -16*x**2 - 16*x - 3 = 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)**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: pass — The solution correctly applies the chain rule via the derivative of arcsine, followed by algebraic simplification. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the chain rule via the derivative of arcsine, followed by algebraic simplification. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: fail (style) 2026-09-26 — Step 2 is labeled 'derivative' but applies the chain rule; the label 'chain' exists in the vocabulary and should be used. Step 3 is labeled 'derivative' but applies the power/constant-multiple rules to a polynomial; 'algebra' or 'power' would be more precise, though 'derivative' is acceptable for basic differentiation. The primary defect is the mislabeling of the chain rule application in step 2.gpt-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.