If $f(x)=|x-5|+|x+5|+|x-4|+|x+4|$,then $\frac{f^{\prime}(1)-f^{\prime}(-6)}{f^{\prime}(-1)+f^{\prime}(6)}=$

If $f: R \rightarrow R$ is defined by $f(x) = \begin{cases} \frac{x-2}{x^2-3x+2} & \text{if } x \in R - \{1, 2\} \\ 2 & \text{if } x = 1 \\ 1 & \text{if } x = 2 \end{cases}$,then find $\lim_{x \rightarrow 2} \frac{f(x)-f(2)}{x-2}$.

If $y = e^x \log x$,then $\frac{dy}{dx}$ is

If $f(x) = \cos^{-1}\left[ \frac{1 - (\log x)^2}{1 + (\log x)^2} \right]$,then the value of $f'(e)$ is:

If $y = \frac{(1 - x)^2}{x^2}$,then $\frac{dy}{dx}$ is

$\frac{d}{dx}(e^{x^3})$ is equal to