If $x = at^2$ and $y = 2at$, then the value of $\frac{d^2y}{dx^2}$ is

( where t is a parameter )

If $\cos y = x\cos(a+y)$, then $\frac{dy}{dx} =$

If $f(x)=\begin{cases}\frac{\sin(a-b)x}{x}, & x<0\\ 2a+b, & x=0\\ \frac{-\tan 12x}{bx}, & x>0\end{cases}$, $(b>0)$ is continuous at $x=0$, then

If $y = (\log_e x)^{\log_e x}$, then $\dfrac{dy}{dx}$ is equal to:

If $y = \log_e(\sec e^{x^2})$ then $\frac{dy}{dx} =$

The function $f(x) = \begin{cases} \frac{(\sin 2x)}{x} + \cos x & , if \ x \neq 0 \\ K & , if \ x = 0 \end{cases}$ is continuous at $x = 0$, then the value of K is:

If $x = a(\theta+\sin\theta)$, $y = a(1-\cos\theta)$ then $\dfrac{d^2y}{dx^2}$ is equal to: