Mathematics - Differentiation Question with Solution | TestHub

$x=\frac{1}{2},y=\frac{-1}{4}\Rightarrow xy=\frac{-1}{8}$

$y.\frac{\left(-2x\right)}{2\sqrt{1-x^2}}+y^{\text{'}}.\sqrt{1-x^2}=0-\left\{1.\sqrt{1-y^2}+\frac{x.\left(-2y\right)}{2\sqrt{1-y^2}}{y}^{\text{'}}\right\}$

$-\frac{xy}{\sqrt{1-x^2}}+y^{\text{'}}\sqrt{1-x^2}=-\sqrt{1-y^2}+\frac{xy.y^{\text{'}}}{\sqrt{1-y^2}}$

$y^{\text{'}}\left(\sqrt{1-x^2}-\frac{xy}{\sqrt{1-y^2}}\right)=\frac{xy}{\sqrt{1-x^2}}-\sqrt{1-y^2}$

$y^{\text{'}}\left(\frac{\sqrt{3}}{2}+\frac{1}{8.\frac{\sqrt{15}}{4}}\right)=\frac{-1}{8.\sqrt{\frac{3}{4}}}-\frac{\sqrt{15}}{4}$

$y^{\text{'}}\left(\frac{\sqrt{45}+1}{2\sqrt{15}}\right)=-\frac{\left(1+\sqrt{45}\right)}{4\sqrt{3}}$

$y^{\text{'}}=-\frac{\sqrt{15}}{2\sqrt{3}}=-\frac{\sqrt{5}}{2}$