For the function $f(x) = \min\{|x|, |x-2|\}$ which of the following are TRUE?

A. Function $f(x)$ is continous on $\mathbb{R}$

B. Function $f(x)$ is differentible on $\mathbb{R}$

C. Function $f(x)$ is differentible on $\mathbb{R} \sim \{0,1,2\}$

D. Function is continous on $\mathbb{R}$ but not differentiable at $x = \dfrac{1}{2}$

Choose the correct answer from the options given below:

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

If $y = x^{(x \sin x)}$ then $\frac{dy}{dx} = ?$

If $t=e^{2 x}$ and $y=\log _{e} t^{2}$, then $\frac{d^{2} y}{d x^{2}}$ is :

Let $f(x)=\begin{cases}\frac{1-\cos4x}{x^2}, & \text{if } x<0\\2\lambda, & \text{if } x=0\\\frac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4}, & \text{if } x>0\end{cases}$ such that $f$ continuous at $x=0$. Then, the value of $\lambda$ is equal to:

If $y = \frac{1}{\sqrt[3]{1-x^3}}$ then $\frac{dy}{dx}$ is equal to

Derivative of $x^x$ with respect to $x\log x$ is