f = Sum[y[i] Log[Sum[y[i], {i, 1, n}]/(
    1 - Product[(1 - b)^Exp[x[i] [Beta]], {i, 1, n}])], {i, 1, n}];
D[f /. n -> 2, b]

-(((1 - b)^(-1 + E^([Beta] x[1]) +
E^([Beta] x[2])) (E^([Beta] x[1]) + E^([Beta] x[2])) y[1])/(
1 - (1 - b)^(E^([Beta] x[1]) + E^([Beta] x[2])))) - ((1 - b)^(-1 +
E^([Beta] x[1]) + E^([Beta] x[2])) (E^([Beta] x[1])
+ E^([Beta] x[2])) y[2])/(1 - (1 - b)^(E^([Beta] x[1]) + E^([Beta] x[2])))