Question

If \( y = \tan^{-1}\left(\frac{3x - x^{3}}{1 - 3x^{2}}\right) \), then \( \frac{dy}{dx} \) is:

• A. $\frac{3}{1+x^{2}}$
• B. $\frac{1}{1+x^{2}}$
• C. $\frac{3}{1+9x^{2}}$
• D. $\frac{3x^{2}}{1+x^{2}}$

Hint:

Substitution simplifies inverse trigonometric derivatives significantly.

Answer 1

The Correct Option is A

Step 1: Concept
Identify the trigonometric substitution. $\tan 3\theta = \frac{3\tan\theta - \tan^3\theta}{1 - 3\tan^2\theta}$.
Step 2: Analysis
Put $x = \tan \theta$, then $y = \tan^{-1}(\tan 3\theta) = 3\theta = 3 \tan^{-1} x$.
Step 3: Conclusion
Differentiating with respect to $x$, $\frac{dy}{dx} = 3 \times \frac{1}{1+x^{2}}$.
Final Answer: (A)