Will the following be sufficient? It has the derivative term, multiplied by the result of the summation where the limits have been parametrized.
In
Clear["Global`"]; ( Clear. *)
s[x_] := Sum[ k Exp[-a x^3], {k, m, n}]; (* Define the expression. *)
D[ s[x], x] (* Find Derivative. *)
Out[140]= 3/2 a E^(-a x^3) (-1 + m - n) (m + n) x^2
When m and n are assigned values, we will get back a single expression.
In[141]:= % /. {m -> 1, n -> 7}
Out[141]= -84 a E^(-a x^3) x^2