Consider the function $f(x)=\frac{2}{\sqrt[\phantom{2}]{x}}.$

$($a$)$ For $a>0,$ form the difference quotient

$\frac{f(ah)-f(a)}{h}$

and simplify this difference quotient to obtain an expression which may be evaluated at $h=0.$

$($b$)$ Use the definition of the derivative $($see Definition $2\mathrm{.}2\mathrm{.}1$ in CLPI$)$ to find $f^{\text{'}}(a)$ for $a>0,$

$f^{\text{'}}(a)=\underset{h0}{lim}\frac{f(ah)-f(a)}{h}$

$($c$)$ Find the tangent line to the graph of $f$ at $x=9.$

$($d$)$ Find the inverse function $f^{-1}$ and give the domain and range of $f^{-1}.$

$($e$)$ Use the definition of the the derivative to find the derivative of $f^{-1}$ at $\frac{2}{3}.($Note that $f(9)=\frac{2}{3}.)$

$($f$)$ How are the derivatives $f^{\text{'}}(9)$ and $(f^{-1}{)}^{\text{'}}(\frac{2}{3})$ related?