Question

second derivative of arcsin(x)^(1/2)

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{y}={\sin }^{-1}\left(\sqrt{\mathrm{x}}\right) \\ \mathrm{So}\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}{\sin }^{-1}\left(\sqrt{\mathrm{x}}\right)=\frac{1}{\sqrt{1-{\left(\sqrt{\mathrm{x}}\right)}^2}}\times \frac{\mathrm{d}}{\mathrm{dx}}\left(\sqrt{\mathrm{x}}\right) \\ =\frac{1}{\sqrt{1-\mathrm{x}}}\times \frac{\mathrm{d}}{\mathrm{dx}}{\mathrm{x}}^{\frac{1}{2}} \\ =\frac{1}{\sqrt{1-\mathrm{x}}}\times \frac{1}{2\sqrt{\mathrm{x}}}=\frac{1}{2\sqrt{\mathrm{x}-{\mathrm{x}}^2}} \\ \mathrm{Again}\mathrm{differentiating},\mathrm{we}\mathrm{get} \\ \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}=\frac{1}{2}\left(\frac{\mathrm{d}}{\mathrm{dx}}{\left(\mathrm{x}-{\mathrm{x}}^2\right)}^{\frac{-1}{2}}\right)=\frac{1}{2}\times \frac{-1}{2}{\left(\mathrm{x}-{\mathrm{x}}^2\right)}^{\frac{-1}{2}-1}\times \frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}-{\mathrm{x}}^2\right) \\ =\frac{-1}{4}{\left(\mathrm{x}-{\mathrm{x}}^2\right)}^{\frac{-3}{2}}\times \left(1-2\mathrm{x}\right) \end{gathered}$$