If $f:R \to R$ is a differentiable function and $f(1) = 4$,then the value of $\lim_{x \to 1} \int_4^{f(x)} \frac{2t}{x - 1} dt$ is:

For some constants $a$ and $b$,find the derivative of $(x-a)(x-b)$.

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

If $y=\log \left[\tan \sqrt{\frac{2^x-1}{2^x+1}}\right], x>0$, then $\left(\frac{d y}{d x}\right)_{x=1}=$

If $f(x)$ is an even function and $f^{\prime}(x)$ exists,then $f^{\prime}(e) + f^{\prime}(-e)$ is

For $x > 0$ and $(x \log x) < 1$, if $y = \cot^{-1} \left( \frac{x - \log x}{x^2 \log_e x^2 + \log x^x} \right)$, then $\frac{dy}{dx} = \dots$