Find the Missing Letter

C# (.NET Core), 48 47 46 bytes, input as char array

s=>{for(int i=0;s[++i]==++s[0];);return s[0];}

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Explanation: the first element in the array is incremented as well as a pointer iterating the following elements. When both the first element and the current element are different, it returns the first element.

C# (.NET Core), 58 56 50 bytes, input as string

s=>{var c=s[0];while(s.IndexOf(++c)>=0);return c;}

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Previous 58-byte solution (referenced in the first comment):

s=>{for(int i=1;;i++)if(s[i]-s[0]>i)return(char)(s[i]-1);}

Algorithms using System.Linq

The following algorithms must add using System.Linq; (18 bytes) to the byte count and therefore are longer.

I quite liked this one (52+18 bytes):

s=>{int i=0;return(char)(s.First(c=>c-s[0]>i++)-1);}

And you also have a one-liner (45+18)-byte solution:

s=>(char)(s.Where((c,i)=>c-s[0]>i).First()-1)

And a very clever (37+18)-byte solution, courtesy of Ed'ka:

s=>s.Select(e=>++e).Except(s).First()

Alice, 10 bytes

/X.
\ior@/

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Explanation

This is just a framework for linear programs that operate entirely in Ordinal (string processing) mode:

/...
\.../

The actual linear code is then:

i.rXo@

Which does:

i   Read all input.
.   Duplicate.
r   Range expansion. If adjacent letters don't have adjacent code points, the
    intermediate code points are filled in between them. E.g. "ae" would turn
    into "abcde". For the inputs in this challenge, this will simply insert
    the missing letter.
X   Symmetric set difference. Drops all the letters that appear in both strings,
    i.e. everything except the one that was inserted by the range expansion.
o   Output the result.
@   Terminate the program.

Haskell, 33 30 bytes

f a=until(`notElem`a)succ$a!!0

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