Limits, Continuity & Differentiability
Derivatives of Inverse Functions
Grade 12
Question:
<p><span>\(\frac{d}{dx}\left[\sin^{-1}\left(x\sqrt{1-x} - \sqrt{x}\sqrt{1-x^2}\right)\right]\)</span> is equal to</p>
<p>(a) <span>\(-1\)</span></p>
<p>(b) <span>\(\frac{1}{2}\)</span></p>
<p>(c) <span>\(-\frac{1}{2}\)</span></p>
<p>(d) <span>\(1\)</span></p>
Step-by-Step Solution
Key Concept: Recognizing the argument as a standard inverse sine form after algebraic simplification helps.
<p><strong>Solution:</strong> Simplify the argument using trigonometric substitution. Let <span>$x = \sin^2 \theta$</span>. The expression simplifies and the derivative evaluates to <span>$1$</span>.</p>
Correct Answer: d