The derivative of $\sin(\tan^{-1} e^{2x})$ with respect to $x$ is:

The given function $f(x) = [x]$ is discontinuous at :

If $y = 3e^{2x} + 2e^{3x}$, then $\frac{d^2y}{dx^2} + 6y$ is equal to

For the function $f(x) = -2|\log_e|x||$, which of the following is true?

For what value of $\alpha$, the function $f$ defined by $f(x) = \begin{cases} \alpha(x^2 - 2x + 1), & \text{if } x \leq 0 \\ 2x + 1, & \text{if } x > 0 \end{cases}$ is continuous at $x = 0$?

If $x = 3t^2$ and $y = \sqrt{t}$, then the value of $\frac{d^2y}{dx^2}$ is

If $y = 500 e^{7x} + 600 e^{-7x}$ and $\frac{d^2 y}{dx^2} = ky$, then the value of k is :