Let $f(x)=\frac{1}{2x}$

$($a$)$ Find $f^{\text{'}}(x).$

$($b$)$ Find the slope of the tangent line to the graph of $f$ at the point where $x=1.$

$($c$)$ sketch the graph of $f$ and the tangent line to the curve at the point found in part $($b$).$

$($d$)$ What is the rate of change of $f$ at this point?