Home›Calculus 1›Derivatives of inverse trig functions›Problem 2.832

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 \( \frac{2}{\sqrt{1 - \left(2 x + 4\right)^{2}}} \)

✓ 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. How we verify

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 - (2x + 4)*2 = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 1 - (2x + 4)*2 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 1 - (2x + 4)*2 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 1 - (2x + 4)*2 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 1 - (2x + 4)2 = 0 undefined where -4x*2 - 16x - 15 = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where -4x2 - 16x - 15 = 0
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0 undefined where 1 - (2x + 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: pass
qwen3.6:27b-mlx: fail (error) — Step 4 applies two rules at once: it differentiates the linear term (derivative rule) and evaluates the constant derivative (constant rule) in a single step, violating the one-change-per-step constraint.

Senior review claude-sonnet-5-5, 2026-10-13: pass — The arcsine rule with the chain factor, the sum rule, evaluation of the simple derivatives, and the algebraic expansion are all correct, and every line was proved equivalent. The final line equals the stated answer.

qwen3.6:27b-mlx: dismiss — Step 4 evaluates the two trivial derivatives (d/dx 2x = 2 and d/dx 4 = 0) left by the sum rule in one pass, which is a single evaluation step, and the 'constant' label names a rule it really applies.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-13 — The arcsine rule with the chain factor, the sum rule, evaluation of the simple derivatives, and the algebraic expansion are all correct, and every line was proved equivalent. The final line equals the stated answer.
qwen3.6:27b-mlx: fail (error) 2026-09-27 — Step 4 applies two rules at once: it differentiates the linear term (derivative rule) and evaluates the constant derivative (constant rule) in a single step, violating the one-change-per-step constraint.
gpt-oss:20b: pass 2026-09-27
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 2026-09-26
qwen3.6:27b-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 2026-09-27 with SymPy 1.14.0.