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I'm having difficulties typing the following:

What I am getting with the code

beginalignat*4
 f(x)&=x^-1, &&qquad&& f(2) &&= 2^-1=frac12,\
 f'(x)&=-x^-2, &&qquad &&f'(2) &&=-frac12^2,\
 f''(x)&=2!x^-3, &&qquad &&fracf''(2)2!&&=2^-3=frac12^3,\
 f'''(x)&=-3!x^-4, &&qquad &&fracf'''(2)3!&&=-frac12^4,\
 &vdotswithin=&&qquad &&vdotswithin=&&\
 f^(n)(x)&=(-1)^nn!x^-(n+1), &&qquad&& fracf^(n)(2)n!&&=frac(-1)^n2^n+1.
 endalignat*

is something like this:

I have tried very hard but cannot get the spacing right. Can someone please help me with this?

OK, here we go. A simple align (of course with the appropriately set &) yields

documentclassarticle
usepackagemathtools
begindocument
beginalign
 f(x)&=x^-1, & f(2) &= 2^-1=frac12,\
 f'(x)&=-x^-2, &f'(2) &=-frac12^2,\
 f''(x)&=2!x^-3, &fracf''(2)2!&=2^-3=frac12^3,\
 f'''(x)&=-3!x^-4, &fracf'''(2)3!&=-frac12^4,\
 &vdotswithin= & &vdotswithin=\
 f^(n)(x)&=(-1)^nn!x^-(n+1), & fracf^(n)(2)n!&=frac(-1)^n2^n+1
endalign
enddocument 

You have too many &'s. Just three per row suffice.

On the other hand, aligning at the = signs doesn't seem the best idea, as it creates irregular shapes and the = signs are not really related to one another, so I propose a different solution.

I also added , after the ! when the factorial is immediately followed by another term.

documentclassarticle
usepackagemathtools

begindocument

beginalignat*2
f(x)&=x^-1, qquad & f(2) &= 2^-1=frac12,
\
f'(x)&=-x^-2, qquad & f'(2) &=-frac12^2,
\
f''(x)&=2!,x^-3, qquad & fracf''(2)2!&=2^-3=frac12^3,
\
f'''(x)&=-3!,x^-4, qquad & fracf'''(2)3!&=-frac12^4,
\
&vdotswithin= qquad & & vdotswithin=
\
f^(n)(x)&=(-1)^nn!,x^-(n+1), qquad & fracf^(n)(2)n!&=frac(-1)^n2^n+1.
endalignat*

beginalignat*2
& f(x)=x^-1, && f(2) = 2^-1=frac12,
\[1ex]
& f'(x)=-x^-2, && f'(2) =-frac12^2,
\[1ex]
& f''(x)=2!,x^-3, && fracf''(2)2!=2^-3=frac12^3,
\[1ex]
& f'''(x)=-3!,x^-4, && fracf'''(2)3!=-frac12^4,
\
&qquadvdots && qquadvdots
\
& f^(n)(x)=(-1)^nn!,x^-(n+1),qquad && fracf^(n)(2)n!=frac(-1)^n2^n+1.
endalignat*

enddocument