Using the quotient rule,
$$ f(x)=\frac{x}{\tan{x}}$$
$$ f'(x)=\frac{\tan{x}(1)-x(\sec^2{x})}{\tan^2{x}} $$
$$ = \frac{\tan{x}-x\sec^2{x}}{\tan^2{x}}$$
Using our knowledge of the unit circle,
$$ \tan{\frac{\pi}{4}}=1 $$
$$ \sec{\frac{\pi}{4}}=\frac{1}{\cos{\frac{\pi}{4}}} = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}} $$
$$ \sec^2{\left(\frac{\pi}{4}\right)}=\left(\frac{2}{\sqrt2}\right)^2=\frac{4}{2}=2$$
Substituting our values,
$$f'(x) = \frac{\tan{x}-x\sec^2{x}}{\tan^2{x}}$$
$$f'\left(\frac{\pi}{4}\right) = \frac{\tan{(\frac{\pi}{4})}-\frac{\pi}{4}\sec^2{(\frac{\pi}{4}})}{\tan^2{\left(\frac{\pi}{4}\right)}}$$
$$ = \frac{1-\frac{\pi}{4}(2)}{(1)^2}$$
$$ = \boxed{1-\frac{\pi}{2}}$$