Derivative of $\log (\sec \theta+\tan \theta)$ with respect to $\sec \theta$ at $\theta=\frac{\pi}{4}$ is

Find the slope of the normal to the curve $x=a \cos ^{3} \theta, y=a \sin ^{3} \theta$ at $\theta=\frac{\pi}{4}$.

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 $y = \sin \theta - \sin 2\theta$.

If $x = a \cos \theta$ and $y = a \sin \theta$,then $\frac{d^2 y}{d x^2} = \rule{1cm}{0.15mm} \, (a \neq 0; \theta \neq k\pi, k \in Z)$.

The $2^{nd}$ derivative of $a \sin^3 t$ with respect to $a \cos^3 t$ at $t = \frac{\pi}{4}$ is

If $x = \sin \theta$ and $y = \sin^3 \theta$,then the value of $\frac{d^2 y}{d x^2}$ at $\theta = \frac{\pi}{6}$ is: