The convolution operation is stated below
and is equivalent to
Now, lets say we have 2 functions namely x(t) and u1(t). If we convolve the x(t) with u1(t) where u1(t) is the unit doublet function, the behavior of the output becomes
x'(t) = x(t) ※ u1(t)
we can seee that everytime we convolve an input function (which is x(t)) with the unit doublet, the output is the derivative of the input function which is x'(t).
Now, according here
The question remains, why is that the integrals above are equated to -g'(t) and not g'(-t)
Reference: http://isites.harvard.edu/fs/docs/icb.topic133182.files/5-singular_func.pdf page 4, number 2