Question

If $y=\sin \left(2{\sin }^{-1}x\right)$, then prove that: $\frac{dy}{dx}=2\sqrt{\frac{1-y^2}{1-x^2}}$.

Answer 1

Given that $y=\sin \left(2{\sin }^{-1}x\right)$

$\Rightarrow {\sin }^{-1}y=2{\sin }^{-1}x$

Differentiation with respect to $x$.

$\therefore \frac{1}{\sqrt{1-y}}\frac{dy}{dx}=2\frac{1}{\sqrt{1-x^2}}$

$\frac{dy}{dx}=2\sqrt{\frac{1-y^2}{1-x^2}}$. $\left[henceproved\right]$