If $x=2at^5$ and $y=at^{10}$, then $\dfrac{dy}{dx}=$

If $y = \left(\log\left(x + \sqrt{x^2+a^2}\right)\right)^2$ and $x \neq \frac{1-a^2}{2}$, then $(x^2+a^2)\frac{d^2y}{dx^2} + x\frac{dy}{dx}$ is equal to:

A function $f(x) = \begin{cases} \frac{kx^3}{4} & \text{if } x \leq 1 \\ 8 & \text{if } x > 1 \end{cases}$ for some constant $k$, then $\frac{d^2f(x)}{dx^2}$ at $x = -1$ is

The function $f: \mathbb{R} \to \mathbb{R}$ defined by

$f(x) = \begin{cases} x^2, & x \ge 1 \\ x, & x < 1 \end{cases}$

is

If $y = \sqrt{12^x(3x^2+2)}$, then $\dfrac{dy}{dx}$ will be

(consider $\log_e x = \log x$)

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

If $y = \left(x + \sqrt{x^2+1}\right)^m$, then $\frac{dy}{dx}$ is