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I am not sure to understand your request, but if it is OK to work with n fixed, does the following help you ?
sage: n = 3
....: var('a','b')
....: assume(a>0,b>0)
....: X = [SR.var('X_{}'.format(i)) for i in range(n)]
....: L = prod(a*b*X[i]^(a-1)*(1-X[i]^a)^(b-1) for i in range(n))
sage: L
X_0^(a - 1)*(-X_0^a + 1)^(b - 1)*X_1^(a - 1)*(-X_1^a + 1)^(b - 1)*X_2^(a - 1)*(-X_2^a + 1)^(b - 1)*a^3*b^3
sage: L.diff(a)
-X_0^(a - 1)*X_0^a*(-X_0^a + 1)^(b - 2)*X_1^(a - 1)*(-X_1^a + 1)^(b - 1)*X_2^(a - 1)*(-X_2^a + 1)^(b - 1)*a^3*(b - 1)*b^3*log(X_0) - X_0^(a - 1)*(-X_0^a + 1)^(b - 1)*X_1^(a - 1)*X_1^a*(-X_1^a + 1)^(b - 2)*X_2^(a - 1)*(-X_2^a + 1)^(b - 1)*a^3*(b - 1)*b^3*log(X_1) - X_0^(a - 1)*(-X_0^a + 1)^(b - 1)*X_1^(a - 1)*(-X_1^a + 1)^(b - 1)*X_2^(a - 1)*X_2^a*(-X_2^a + 1)^(b - 2)*a^3*(b - 1)*b^3*log(X_2) + X_0^(a - 1)*(-X_0^a + 1)^(b - 1)*X_1^(a - 1)*(-X_1^a + 1)^(b - 1)*X_2^(a - 1)*(-X_2^a + 1)^(b - 1)*a^3*b^3*log(X_0) + X_0^(a - 1)*(-X_0^a + 1)^(b - 1)*X_1^(a - 1)*(-X_1^a + 1)^(b - 1)*X_2^(a - 1)*(-X_2^a + 1)^(b - 1)*a^3*b^3*log(X_1) + X_0^(a - 1)*(-X_0^a + 1)^(b - 1)*X_1^(a - 1)*(-X_1^a + 1)^(b - 1)*X_2^(a - 1)*(-X_2^a + 1)^(b - 1)*a^3*b^3*log(X_2) + 3*X_0^(a - 1)*(-X_0^a + 1)^(b - 1)*X_1^(a - 1)*(-X_1^a + 1)^(b - 1)*X_2^(a - 1)*(-X_2^a + 1)^(b - 1)*a^2*b^3
