DeltaFunctionPrime[1 - x] is the derivative of the Dirac
delta-function \delta (x).
Overview, Convolute, DeltaFunction, DeltaFunctionDoublePrime, Integrate2, SimplifyDeltaFunction.
DeltaFunctionPrime[1 - x]\delta '(1-x)
Integrate2[DeltaFunctionPrime[1 - x] f[x], {x, 0, 1}]f'(1)
Integrate2[DeltaFunctionPrime[1 - x] x^2, {x, 0, 1}]2