This is possibly a duplicate, but the answers given in most cases seem probably more in depth than the OP is looking for, being a newcomer to Mathematica.
First you should define f as a function.
f[e_] := 7.523190091795795`*^-18 (1.329223358440088`*^17 -
3.9876700753202693`*^18 e +
Sqrt[-8.944600854152669`*^34 + 1.4311361366644287`*^37 e^2]);
From there the tangent line to some point is given by first two terms of the series expansion, so to get the plot you're looking for you can use Mathematica's Series
function, which you can read about in the docs.
Plot[{
f[x],
Evaluate@Normal@Series[f[x], {x, 1/4, 1}]
}, {x, 0.1, 0.3}
, PlotRange -> All
]
Here Evaluate@Normal@...
is just preparing the output of Series
for the Plot
function. This give the following image, where the blue line is your original function, and the orange is the tangent:

qlink[n_] := StringReplace["[(qn)](http://mathematica.stackexchange.com/q/qn)", "qn" :> ToString[n]]
$\endgroup$ – Michael E2 Jul 2 '15 at 14:31