if $x=\sqrt{a^{\tan^{-1}t}}$ and $y=\sqrt{a^{\cot^{-1}t}}$ then which one of the following is true?

If $y = \log_e \sin(e^x + 5x + 500)$ then $\dfrac{dy}{dx}$ is equal to

The function $f(x) = e^{|x|}$ is

(a) continuous everywhere on $R$

(b) not continuous at $x = 0$

(c) Differentiable everywhere on $R$

(d) not differentiable at $x = 0$

(e) continuous and differentiable on $R$

Choose the most appropriate answer from the options given below :

If $x = 2\sin\theta$ and $y = 2\cos\theta$, then the value of $\frac{d^2y}{dx^2}$ at $\theta = 0$ is

If $y = \sin^{-1}x$, then $(1-x^2)\frac{d^2y}{dx^2}$ is equal to

If $f(x)$, defined by $f(x)=\left\{\begin{array}{lll}k x+1 & \text { if } & x \leq \pi \\ \cos x & \text { if } & x>\pi\end{array}\right.$ is continuous at $x=\pi$, then the value of $k$ is :

If $y = \frac{1}{x+1}$, then $\frac{d^2y}{dx^2}$ at $x = 2$ is: