Question

\(\frac{d}{dx} \left[ 2 \tan^{-1} x \right] \)

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

Hint:

The derivative of \( \tan^{-1} x \) is \( \frac{1}{1 + x^2} \), and multiplying by a constant gives the final result.

Answer 1

The Correct Option is D

We are asked to differentiate \( 2 \tan^{-1} x \). Step 1: Apply the derivative of inverse tangent
The derivative of \( \tan^{-1} x \) is: \[ \frac{d}{dx} \tan^{-1} x = \frac{1}{1 + x^2} \]
Step 2: Multiply by the constant 2
Since the expression is \( 2 \tan^{-1} x \), we apply the constant multiple rule: \[ \frac{d}{dx} \left( 2 \tan^{-1} x \right) = 2 \cdot \frac{1}{1 + x^2} = \frac{2}{1 + x^2} \]