Derivation of Interpolation function

I have a problem to derive an interpolation function.
I am quite new to mathematica, so I suspect I miss something.

I have a list of values of 1000 values (L) that represents a function that I want to derive.
The list looks like this :

I interpolate the list

InterL = Interpolation[L, InterpolationOrder -> 1]

I verify that the interpolation works :

In[399]:= InterL[400.6]

Out[399]= 686.8

If I plot InterL I have a similar curve as L except that is defined on everypoint.

Now if I try to derive InterL and plot it I get this curve :

Which doesn’t look like the derivative of the precedent. I have tried to evaluate the derivative, etc. I can’t find a solution.

Is it possible to derive this curve ?