If $f(x) = \sum_{p=1}^7 p^2 \sin^{-1}\left(\frac{4}{5} \sin(px) - \frac{3}{5} \cos(px)\right)$, then the value of $\frac{df}{dx}$ at $x = 1$ is (Given that $\sin^{-1}(\sin x) = x$)

If $f(x) = x^2 \sin \frac{1}{x}$ for $x \neq 0$ and $f(0) = 0$,then find $\lim_{x \rightarrow 0} f^{\prime}(x)$.

$\frac{d}{dx} \log(\log x) =$

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} = $

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

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