$($a$)$ Compute f$($x$)=1$xsin$($t$2)2$tdt$.\{\backslash$displaystyle f$($x$)=\backslash$int $\_\{-1\}^\{$x$\}\backslash$sin$($t$^\{2\})2$t$\backslash ,$dt$.\}($b$)$ Find f$($x$).\{\backslash$displaystyle f$\text{'}($x$).\}($c$)$ State the Fundamental Theorem of Calculus.$($d$)$ Use the Fundamental Theorem of Calculus to compute ddx$(1$xsin$($t$2)2$tdt$)\{\backslash$displaystyle $\{\backslash$frac $\{$d$\}\{$dx$\}\}\{\backslash$bigg $(\}\backslash$int $\_\{-1\}^\{$x$\}\backslash$sin$($t$^\{2\})2$t$\backslash ,$dt$\{\backslash$bigg $)\}\}$ without first computing the integral.