Differentiate the following with respect to $x:$ $\sin (\log x), x > 0$

If the function $f(x)$ is defined by $f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + \dots + \frac{x^2}{2} + x + 1$,then $f'(0) = $

Let $f(x)$ be a second degree polynomial. If $f(1) = f(-1)$ and $p, q, r$ are in $A$.$P$.,then $f^{\prime}(p), f^{\prime}(q), f^{\prime}(r)$ are

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 $PROPERTY \ 1$

The differential coefficient of ${x^6}$ with respect to ${x^3}$ is

Differentiate the following with respect to $x$: $\sqrt{e^{\sqrt{x}}}, x > 0$