If $x=\sin heta, y=\sin ^3 heta$, then $\frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}$ at $ heta=\frac{\pi}{2}$ is

Question

If $x=\sin 	heta, y=\sin ^3 	heta$, then $\frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}$ at $	heta=\frac{\pi}{2}$ is

Aicharya · Accepted Answer

$\begin{array}{ll} & x=\sin 	heta 	ext { and } y=\sin ^3 	heta \ 	herefore \quad & y=x^3 \ 	herefore \quad & \frac{\mathrm{~d} y}{\mathrm{~d} x}=3 x^2 \ 	herefore \quad & \frac{\mathrm{~d}^2 y}{\mathrm{~d} x^2}=6 x \ & 	ext { At } 	heta=\frac{\pi}{2}, x=\sin \frac{\pi}{2}=1 \ 	herefore \quad & \left(\frac{\mathrm{~d}^2 y}{\mathrm{~d} x^2}ight)_{	heta=\frac{\pi}{2}}=\left(\frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}ight)_{x=1}=6(1)=6\end{array}$

If $x=\sin \theta, y=\sin ^3 \theta$, then $\frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}$ at…

Solution