Question

The derivative of \( \tan^{-1}(x^2) \) w.r.t. \( x \) is:

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

Hint:

When differentiating inverse trigonometric functions, use the chain rule and simplify the result.

Answer 1

The Correct Option is B

Step 1: Apply the chain rule.
The derivative of \( \tan^{-1}(u) \) is: \[ \frac{d}{dx} \tan^{-1}(u) = \frac{1}{1 + u^2} \cdot \frac{du}{dx}. \] Here, \( u = x^2 \), so \( \frac{du}{dx} = 2x \). Step 2: Substitute and simplify.
\[ \frac{d}{dx} \tan^{-1}(x^2) = \frac{1}{1 + (x^2)^2} \cdot 2x = \frac{2x}{1 + x^4}. \] Step 3: Conclusion.
The derivative is: \[ \boxed{\frac{2x}{1 + x^4}}. \]