If $x = \sin t \cos 2t$ and $y = \cos t \sin 2t$,then at $t = \frac{\pi}{4}$,the value of $\frac{dy}{dx}$ is equal to

If $x = \frac{1 + t}{t^3}$ and $y = \frac{3}{2t^2} + \frac{2}{t}$,then $x \left( \frac{dy}{dx} \right)^3 - \frac{dy}{dx}$ is equal to (where $t$ is a real parameter).

If $y=e^{\sin ^{-1}(t^{2}-1)}$ and $x=e^{\sec ^{-1}(\frac{1}{t^{2}-1})}$,then $\frac{dy}{dx}$ is equal to

If $x=t^2+t+1$ and $y=\sin \left(\frac{t \pi}{2}\right)+\cos \left(\frac{t \pi}{2}\right)$,then find the value of $\frac{dy}{dx}$ at $t=1$.

If $x=\cos \theta$ and $y=\sin 5 \theta$,then $\left(1-x^2\right) \frac{d^2 y}{d x^2}-x \frac{d y}{d x}$ is equal to (in $y$)

If $x$ and $y$ are connected parametrically by the equations,without eliminating the parameter,find $\frac{dy}{dx}$ for $x = \cos \theta - \cos 2\theta$ and $y = \sin \theta - \sin 2\theta$.