derivative of a blured function

Question

to blur a function $f(x)$ you convolve it with $e^{-x^2}$
so the convolution = $f(x) * e^{-x^2}$
then I want to find its derivative so
derivative of convolution = $f(x) * (-2xe^{-x^2})$
derivative of convolution = $\int f(y)(-2(x-y)e^{-(x-y)^{2}})dy$
and $f(x) = e^{ae^{cx+d}+b}$ so the final equation to solve is
$\int e^{ae^{cx+d}+b}(-2(x-y)e^{-(x-y)^{2}})dy$
is there a way to solve or get an aproximate solution?
I want to solve various integrals with longer f(x) but similar
some tips on how to solve this will help me find a general solution to those problems