SimplifyDeltaFunction[exp, x] simplifies f[x]*DeltaFunction[1-x] to Limit[f[x],x->1] DeltaFunction[1-x] and applies a list of transformation rules for DeltaFunctionPrime[1-x]*x^(OPEm-1)*f[x] where x^(OPEm-1) is suppressed in exp.
Overview, DeltaFunction, DeltaFunctionPrime.
g[x] DeltaFunction[1 - x]
SimplifyDeltaFunction[ %, x]g(x) \delta (1-x)
\delta (1-x) \underset{x\to 1}{\text{lim}}g(x)
g[x] DeltaFunctionPrime[1 - x]
SimplifyDeltaFunction[ %, x]
x Log[x] DeltaFunctionPrime[1 - x]
SimplifyDeltaFunction[ %, x]g(x) \delta '(1-x)
\delta (1-x) \underset{x\to 1}{\text{lim}}g'(x)+\delta '(1-x) \underset{x\to 1}{\text{lim}}g(x)
x \log (x) \delta '(1-x)
\delta (1-x)
PolyLog[2, 1 - x] DeltaFunctionPrime[1 - x]
SimplifyDeltaFunction[ %, x]\text{Li}_2(1-x) \delta '(1-x)
-\delta (1-x)
Log[x] PolyLog[2, 1 - x] DeltaFunctionPrime[1 - x]
SimplifyDeltaFunction[ %, x]\text{Li}_2(1-x) \log (x) \delta '(1-x)
0
PolyLog[3, 1 - x] DeltaFunctionPrime[1 - x]
SimplifyDeltaFunction[ %, x]\text{Li}_3(1-x) \delta '(1-x)
-\delta (1-x)