y=cos^(–1)(2xsqrt(1–x²)) হলে  (dy)/(dx)= কত?

Explanation:

Another Explanation (5): দেওয়া আছে, \( y = \cos^{-1}(2x\sqrt{1-x^2}) \) ধরি, \( x = \sin\theta \). সুতরাং, \( \theta = \sin^{-1}x \) তাহলে, \( y = \cos^{-1}(2\sin\theta\sqrt{1-\sin^2\theta}) \) \( = \cos^{-1}(2\sin\theta\cos\theta) \) \( = \cos^{-1}(\sin2\theta) \) \( = \cos^{-1}(\cos(\frac{\pi}{2} - 2\theta)) \) \( = \frac{\pi}{2} - 2\theta \) \( = \frac{\pi}{2} - 2\sin^{-1}x \) এখন, \( x \) এর সাপেক্ষে অন্তরকলন করে পাই, \( \frac{dy}{dx} = \frac{d}{dx}(\frac{\pi}{2} - 2\sin^{-1}x) \) \( = 0 - 2 \cdot \frac{1}{\sqrt{1-x^2}} \) \( = \frac{-2}{\sqrt{1-x^2}} \) অতএব, \( \frac{dy}{dx} = \frac{-2}{\sqrt{1-x^2}} \) 🥳