$(a)(20$ points$)$ Use the definition $of$ the derivative $to$ prove that

$\frac{d}{dx}(\frac{1}{\sqrt[\phantom{2}]{x}})=-\frac{1}{2x^{\frac{3}{2}}}$

$(b)(6$ points$)$ Find the tangent line $tofat$ the point $(4$;$\frac{1}{2}).$