Question

If \( y = \frac{\tan x \cos^{-1}x}{\sqrt{1 - x^2}} \), then the value of \( \frac{dy}{dx} \) when \( x = 0 \) is:

• A. \( 0 \)
• B. \( \frac{\pi}{2} \)
• C. \( 1 \)
• D. \( \frac{\pi}{6} \)

Hint:

Use the quotient rule when differentiating rational functions. - The derivative of \( \cos^{-1}x \) is \( \frac{-1}{\sqrt{1-x^2}} \).

Answer 1

The Correct Option is B

Step 1: Differentiate using quotient rule
Given function: \[ y = \frac{\tan x \cos^{-1}x}{\sqrt{1 - x^2}} \] Let \( u = \tan x \cos^{-1}x \) and \( v = \sqrt{1 - x^2} \). Using the quotient rule: \[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \] Step 2: Compute derivatives
For \( u = \tan x \cos^{-1}x \), use the product rule: \[ \frac{du}{dx} = \sec^2 x \cos^{-1}x + \tan x \left(\frac{-1}{\sqrt{1 - x^2}}\right). \] For \( v = \sqrt{1 - x^2} \), \[ \frac{dv}{dx} = \frac{-x}{\sqrt{1 - x^2}}. \] Step 3: Evaluate at \( x = 0 \)
\[ \frac{dy}{dx} \bigg|_{x=0} = \frac{\frac{\pi}{2} (1) - 0}{1} = \frac{\pi}{2}. \] Thus, the correct answer is \( \boxed{\frac{\pi}{2}} \).