If $x^2y^2 = a^{\pi/2}$, then $\dfrac{d^2y}{dx^2} =$

If $x = 16at^{\frac{1}{4}}$ and $y = 8at^{\frac{3}{4}}$ then $\dfrac{dy}{dx}$ is equal to:

If the function $f(x) = \dfrac{(4^x-1)^3}{\sin\dfrac{x}{4}\cdot\log\left(1+\dfrac{x^2}{3}\right)}$ is continuous at $x = 0$, then the value of $f(0)$ is: (Take $\log_e x = \log x$)

If $y=7^x\log(5x^3)$, then $\dfrac{dy}{dx}$ is equal to: (consider $\log_e x = \log x$)

If $x = \frac{1-t}{1+t}$ and $y = \frac{3t}{1+t}$, then $\frac{d^2y}{dx^2}$ is equal to

If a function $f(x) = \begin{cases}\lambda(x^2-2x), & x \le 0\\4x+1, & x>0\end{cases}$ then

If $x = \int_0^y \frac{dt}{\sqrt{1+9t^2}}$ and $\frac{d^2y}{dx^2} = \lambda y$, then, $\lambda$ is equal to