If $y=\sqrt[5]{\log_e(5x^2)}$, then $\dfrac{dy}{dx}=$

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

The value of k for which the function, defined by, $f(x) = \begin{cases} \frac{3x + 4 \tan x}{x} & : x \neq 0 \\ k & : x = 0 \end{cases}$ is continuous at $x = 0$, is

The derivative of $\sin^{-1}\left(\frac{2x}{1+x^2}\right)$ w.r.t. $\tan^{-1}\left(\frac{2x}{1-x^2}\right)$ is

If $e^y(x+1)=1$, then

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

If the function defined by $f(x) = \begin{cases} \ kx^2 + 1, & \text{if } x \le 1 \\ 2 , & \text{if } x > 1 \end{cases}$ is continuous at $x = 1$, then k is equal to

For the function $y(x)=\sqrt{x+1}-\sqrt{x-1}$, which of the following are true?

(A) $\left(\frac{dy}{dx}\right)_{at\,x=2}=\frac{1-\sqrt{3}}{2\sqrt{3}}$

(B) $(x^2-1)\frac{d^2y}{dx^2}-\frac{dy}{dx}+4y=0$

(C) $(x^2-1)\left(\frac{dy}{dx}\right)^2-4y^2=0$

(D) If $P(x)\frac{d^2y}{dx^2}+Q(x)\frac{dy}{dx}+\alpha y=0$, then $P(x)-xQ(x)-4\alpha=0$

Choose the correct answer from the options given below: