If $y = \log \tan \sqrt{x}$,then the value of $\frac{dy}{dx}$ is

People living on Mars,instead of the usual definition of derivative $D f(x)$,define a new kind of derivative,$D^*f(x)$ by the formula $D^*f(x) = \lim_{h \to 0} \frac{f^2(x + h) - f^2(x)}{h}$ where $f^2(x)$ means $[f(x)]^2$. If $f(x) = x \ln x$,then the value of $\left. D^*f(x) \right|_{x = e}$ is:

Let $f: R \rightarrow R$ be a function such that $f(x)=x^3+x^2 f^{\prime}(1)+x f^{\prime \prime}(2)+6, x \in R$,then $f(2)$ equals

If $y = \frac{e^{2x} + e^{-2x}}{e^{2x} - e^{-2x}}$,then $\frac{dy}{dx} = $

Find the derivative of $x^{-3}(5+3x)$.

The differential coefficient of $a^x + \log x \cdot \sin x$ is