Question

If \(y = \sin\left(\tan^{-1}\left(\frac{1}{\sqrt{x^2 - 1}}\right)\right), x > 1\), then \(\frac{dy}{dx} =\)

• A. \(\frac{1}{x^2}\)
• B. \(\frac{1}{x^4}\)
• C. \(\frac{-1}{x^2}\)
• D. \(\frac{-1}{x^4}\)
• E. \(\frac{1}{x^3}\)

Hint:

Whenever you see \(\sqrt{x^2 - 1}\) in an inverse trig function, substituting \(x = \sec \theta\) is almost always the fastest path to simplification.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We can simplify the expression using inverse trigonometric substitutions before differentiating.
Let \(x = \sec \theta\). Then \(\sqrt{x^2 - 1} = \tan \theta\).

Step 2: Key Formula or Approach:
1. Substitution: \(x = \sec \theta \implies \theta = \sec^{-1} x\).
2. Trigonometric identity: \(\frac{1}{\tan \theta} = \cot \theta\).
3. Identity: \(\tan^{-1}(\cot \theta) = \frac{\pi}{2} - \theta\).

Step 3: Detailed Explanation:
Let \(x = \sec \theta\).
\[ y = \sin\left(\tan^{-1}\left(\frac{1}{\sqrt{\sec^2 \theta - 1}}\right)\right) \]
\[ y = \sin\left(\tan^{-1}\left(\frac{1}{\tan \theta}\right)\right) \]
\[ y = \sin\left(\tan^{-1}(\cot \theta)\right) \]
Since \(\tan^{-1}(\cot \theta) = \frac{\pi}{2} - \theta\):
\[ y = \sin(\frac{\pi}{2} - \theta) = \cos \theta \]
Now, convert back to \(x\). Since \(x = \sec \theta\), we have \(\cos \theta = \frac{1}{x}\).
So, \(y = \frac{1}{x}\).
Differentiate with respect to \(x\):
\[ \frac{dy}{dx} = \frac{d}{dx} \left(\frac{1}{x}\right) = -\frac{1}{x^2} \]

Step 4: Final Answer:
The derivative is \(-\frac{1}{x^2}\).