If $y = \log(\tan(x/2)) + \sin^{-1}(\cos x)$,then $dy/dx$ is

Let $f: R \rightarrow R$ be given by
$f(x) = \begin{cases} x^5+5x^4+10x^3+10x^2+3x+1, & x < 0 \\ x^2-x+1, & 0 \leq x < 1 \\ \frac{2}{3}x^3-4x^2+7x-\frac{8}{3}, & 1 \leq x < 3 \\ (x-2)\log_e(x-2)-x+\frac{10}{3}, & x \geq 3 \end{cases}$
Then which of the following options is/are correct?
$(1)$ $f^{\prime}$ has a local maximum at $x = 1$
$(2)$ $f$ is onto
$(3)$ $f$ is increasing on $(-\infty, 0)$
$(4)$ $f^{\prime}$ is $NOT$ differentiable at $x = 1$

Match each function in List-$I$ to its derivative given in List-$II$.

List-$I$	List-$II$
$(A) \sin ^{-1}\left(\frac{2 x}{1+x^2}\right)$	$(I) \cos x-\sin x$
$(B) \tan ^{-1}\left(\frac{1-x}{1+x}\right)$	$(II) \frac{-1}{1+x^2}$
$(C) e^{\log (\sin x+\cos x)}$	$(III) \frac{2}{1+x^2}$
$(D) \sqrt{1-\sin 2 x} \text{ for } (0 < x < \frac{\pi}{4})$	$(IV) \cos x+\sin x$
	$(V) -\sin x-\cos x$

The correct match is:

Let $y = f(x) = \begin{cases} e^{-\frac{1}{x^2}}, & \text{if } x \neq 0 \\ 0, & \text{if } x = 0 \end{cases}$. Then which of the following can best represent the graph of $y = f(x)$?

If $\alpha$ is the angle of intersection between the curves $y = a^x$ and $y = b^x$,then what is $\tan \alpha$ equal to?

If $y = \frac{\tan x \cos^{-1} x}{\sqrt{1-x^2}}$, then the value of $\frac{dy}{dx}$ when $x = 0$ is: