Question

If $x$ and $y$ are connected parametrically by the equation given exercise without eliminating the parameter, Find $\frac{dy}{dx}$.
$x=\frac{{\sin }^{3}t}{\sqrt{\cos 2t}},y=\frac{{\cos }^{3}t}{\sqrt{\cos 2t}}$

Answer 1

$\frac{dx}{dt}=\frac{{\sin }^{3}t\frac{-\sin 2t}{\sqrt{\cos 2t}}+\sqrt{\cos 2t}\left(3\right){\sin }^{2}t\cos t}{\cos 2t}$

$\frac{dy}{dt}=\frac{{\cos }^{3}t\frac{-\sin 2t}{\sqrt{\cos 2t}}+\sqrt{\cos 2t}\left(3\right){\cos }^{2}t\sin t}{\cos 2t}$

$\frac{dy}{dx}=\frac{-2\sin t{\cos }^{4}t+3\cos 2t{\cos }^{2}t\sin t}{-2{\sin }^{4}t\cos t+3\cos 2t{\sin }^{2}t\cos t}$

$\frac{dy}{dx}=\frac{-2{\cos }^{3}t+3\cos 2t\cos t}{-2{\sin }^{3}t+3\cos 2t\sin t}$