$y = \sin^{- 1}{\left [ \frac{4 \sqrt{x}}{1 + 4 x} \right ]}$ হলে,   $\left ( \frac{dy}{dx} \right )_{\left ( \left ( 4 , 2 \right ) \right.}$ এর মান কত?

প্রদত্ত:

$y=\sin ^{-1}\left(\frac{4 \sqrt{x}}{1+4 x}\right)$

প্রতিস্থাপন

$\begin{array}{c} \sqrt{x}=\tan \theta \Longrightarrow x=\tan ^{2} \theta \\ \frac{4 \sqrt{x}}{1+4 x}=\frac{4 \tan \theta}{1+4 \tan ^{2} \theta}=\frac{2 \sin 2 \theta}{1+\sin ^{2} 2 \theta} \end{array}$

ডেরিভেটিভ

$\frac{d y}{d x}=\frac{1}{\sqrt{1-\left(\frac{4 \sqrt{x}}{1+4 x}\right)^{2}}} \cdot \frac{d}{d x}\left(\frac{4 \sqrt{x}}{1+4 x}\right)$

ভগ্নাংশের ডেরিভেটিভ

$\frac{d}{d x}\left(\frac{4 \sqrt{x}}{1+4 x}\right)=\frac{(1+4 x) \cdot \frac{2}{\sqrt{x}}-4 \sqrt{x} \cdot 4}{(1+4 x)^{2}}=\frac{2(1+4 x)-16 x}{\sqrt{x}(1+4 x)^{2}}=\frac{2-8 x}{\sqrt{x}(1+4 x)^{2}}$

বিন্দুতে মান $(x=4)$

$\left.\frac{d y}{d x}\right|_{x-4}=\frac{1}{\sqrt{1-\left(\frac{8}{17}\right)^{2}}} \cdot \frac{2-32}{2 \cdot 289}=\frac{17}{15} \cdot \frac{-30}{578}=-\frac{1}{17}$