Derivative of \( \displaystyle \operatorname{asin}{\left(2 x + 4 \right)} \)
Problem 2.832 · easy
Differentiate \( \displaystyle f(x) = \operatorname{asin}{\left(2 x + 4 \right)} \).
- \[ \frac{d}{d x} \operatorname{asin}{\left(2 x + 4 \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(2 x + 4\right)}{\sqrt{1 - \left(2 x + 4\right)^{2}}} \]inverse-trigApply the derivative rule for arcsine.✓ Proved
- \[ = \frac{\frac{d}{d x} 4 + \frac{d}{d x} 2 x}{\sqrt{1 - \left(2 x + 4\right)^{2}}} \]sumDifferentiate the sum inside the parentheses.✓ Proved
- \[ = \frac{2}{\sqrt{1 - \left(2 x + 4\right)^{2}}} \]constant simplifyEvaluate the derivatives of the linear and constant terms. Simplify the expression.✓ Proved
- \[ = \frac{2}{\sqrt{- 4 x^{2} - 16 x - 15}} \]algebra simplifyExpand the squared binomial. Simplify the expression under the square root.✓ Proved
Answer \( \frac{2}{\sqrt{1 - \left(2 x + 4\right)^{2}}} \)
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] undefined where 1 - (2*x + 4)**2 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - (2*x + 4)**2 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - (2*x + 4)**2 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - (2*x + 4)**2 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 - (2*x + 4)**2 = 0 undefined where -4*x**2 - 16*x - 15 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where -4*x**2 - 16*x - 15 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where 1 - (2*x + 4)**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 (error) — Step 4 applies two rules at once: it evaluates the derivative of 2*x (which requires the constant-multiple rule) and the derivative of 4 (constant rule) in a single step, violating the one-rule-per-step constraint.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 4 applies two rules at once: it evaluates the derivative of 2*x (which requires the constant-multiple rule) and the derivative of 4 (constant rule) in a single step, violating the one-rule-per-step constraint.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 4 claims to evaluate the derivatives of both the linear and constant terms, but only applies the constant rule to the term 4. The evaluation of Derivative(2*x, x) to 2 requires the power rule (or derivative rule), which is not labeled. This violates the one-rule-per-step constraint.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.