If a > 0, what is the derivative of arcsin(x/a) with respect to x?

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Multiple Choice

If a > 0, what is the derivative of arcsin(x/a) with respect to x?

Explanation:
Differentiating arcsin of a function uses the rule d/dx [arcsin(u)] = u' / sqrt(1 - u^2). Here u is x/a, so u' = 1/a. Plugging in gives (1/a) / sqrt(1 - (x/a)^2). Simplify the square root: sqrt(1 - x^2/a^2) = sqrt((a^2 - x^2)/a^2) = sqrt(a^2 - x^2)/|a|. Since a > 0, |a| = a, and the expression becomes (1/a) / (sqrt(a^2 - x^2)/a) = 1 / sqrt(a^2 - x^2). So the derivative is 1 / sqrt(a^2 - x^2) (valid for |x| ≤ a).

Differentiating arcsin of a function uses the rule d/dx [arcsin(u)] = u' / sqrt(1 - u^2). Here u is x/a, so u' = 1/a. Plugging in gives (1/a) / sqrt(1 - (x/a)^2). Simplify the square root: sqrt(1 - x^2/a^2) = sqrt((a^2 - x^2)/a^2) = sqrt(a^2 - x^2)/|a|. Since a > 0, |a| = a, and the expression becomes (1/a) / (sqrt(a^2 - x^2)/a) = 1 / sqrt(a^2 - x^2).

So the derivative is 1 / sqrt(a^2 - x^2) (valid for |x| ≤ a).

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