f = (1 + Exp[-(Sqrt[x*x + M*M] - mu)/T])

CForm[D[f, M]]

-((Power(E,(mu - Sqrt(Power(M,2) + Power(x,2)))/T)*M)/
 (T*Sqrt(Power(M,2) + Power(x,2))))

ClearAll[foo]
foo = RawBoxes[Replace[ToBoxes@#, InterpretationBox[a_, b_, c___] :>
      With[{aa = StringReplace[a,
            { "Sqrt" -> "sqrt", "Power(E," -> "exp(", "Power" -> "pow"}]}, aa],
      {0, Infinity}]] &;

foo@CForm[D[f, M]]

Unprotect[Power];
Format[Power[E, a_], CForm] := exp[a]
Format[Power[a_, 1/2], CForm] := sqrt[a]
Format[Power[a_, b_], CForm] := pow[a, b]
Protect[Power];

D[f, M] //CForm