If $x=a \cos^{3} \theta$ and $y=a \sin^{3} \theta$,then $1+\left(\frac{dy}{dx}\right)^{2}$ is

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

If $x = 2 \sqrt{2} \sqrt{\cos 2 \theta}$ and $y = 2 \sqrt{2} \sqrt{\sin 2 \theta}$, $0 < \theta < \frac{\pi}{4}$, then the value of $\frac{dy}{dx}$ at $\theta = 22 \frac{1}{2}^{\circ}$ is

If $x=a \cos ^{3} \theta$ and $y=a \sin ^{3} \theta$,then $\frac{d y}{d x}$ is

If $x = \log t$ and $y + 1 = \frac{1}{t}$,then $e^{-x} \frac{d^{2} x}{d y^{2}} + \frac{d x}{d y} = $

If $\tan u=\sqrt{\frac{1-x}{1+x}}$ and $\cos v=4 x^{3}-3 x$,then $\frac{d u}{d v}=$