$\frac{d}{dx}(\sin 2x^2)$ equals

If $y=\tan ^{-1}(\sin \sqrt{x})+\operatorname{cosec}^{-1}\left(e^{2 x+1}\right)$,then $\frac{d y}{d x}=$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function. We say that $f$ has $PROPERTY \ 1$ if $\lim_{h \rightarrow 0} \frac{f(h)-f(0)}{\sqrt{|h|}}$ exists and is finite,and $PROPERTY \ 2$ if $\lim_{h \rightarrow 0} \frac{f(h)-f(0)}{h^2}$ exists and is finite. Then which of the following options is/are correct?
$(1) \ f(x)=x|x|$ has $PROPERTY \ 2$
$(2) \ f(x)=x^{2/3}$ has $PROPERTY \ 1$
$(3) \ f(x)=\sin x$ has $PROPERTY \ 2$
$(4) \ f(x)=|x|$ has $PROPERTY \ 1$

If $2f(\sin x) + f(\cos x) = x,$ then $\frac{d}{dx} f(x)$ is

$A$ function $f$,defined for all positive real numbers,satisfies the equation $f(x^2) = x^3$ for every $x > 0$. Then the value of $f'(4) =$

Let $h(x)$ be differentiable for all $x$ and let $f(x) = (kx + e^x) h(x)$ where $k$ is some constant. If $h(0) = 5$,$h'(0) = -2$,and $f'(0) = 18$,then the value of $k$ is equal to