If $x = at^2$ and $y = 2at$, then $\frac{d^2y}{dx^2} = \dots$

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 $x = \frac{1 + t}{t^3}$ 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).

The position of a point at time $t$ is given by $x = a + bt - ct^2$ and $y = at + bt^2$. Its acceleration at time $t$ is:

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

If $u=\sin \left(\frac{x}{y}\right)$,$x=e^t$,and $y=t^2$,then $t^6\left(\frac{d u}{d t}\right)^2 \div \left(e^{2 t}(t-2)^2\right)=$