Compute the derivative of $6 x^{100}-x^{55}+x$.

If $y = \sinh^{-1}\left(\frac{1-x}{1+x}\right)$,then find $\frac{dy}{dx}$.

Let $f$ be any function defined on $R$ and let it satisfy the condition $|f(x) - f(y)| \leq |(x - y)^2|$,for all $(x, y) \in R$. If $f(0) = 1$,then:

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 $y = b \cos \log \left( \frac{x}{n} \right)^n$,then $\frac{dy}{dx} = $

Differentiate the function with respect to $x$: $\frac{\cos^{-1}(\frac{x}{2})}{\sqrt{2x+7}}$,where $-2 < x < 2$.