Mathematics - Differentiation Question with Solution | TestHub

Given, $\begin{array}{rl}y& ={\tan }^{-1}\left(\frac{\sqrt{x}-x}{1+x^{\frac{3}{2}}}\right)\\ y& ={\tan }^{-1}\left(\frac{\sqrt{x}-x}{1+\sqrt{x}\cdot x}\right)\\ y& ={\tan }^{-1}(\sqrt{x})-{\tan }^{-1}x\\ & \left[\because {\tan }^{-1}\left(\frac{x-y}{1+xy}\right)={\tan }^{-1}x-{\tan }^{-1}y\right]\end{array}$ Differentiating w.r.t $x$ $y^{\prime }(x)=\frac{dy}{dx}=\frac{1}{1+x}\times \frac{1}{2\sqrt{x}}-\frac{1}{1+x^2}$ Now, $y^{\prime }(1)=\frac{1}{2}\times \frac{1}{2}-\frac{1}{2}=-\frac{1}{4}$