Differentiate the function f(x) = (3-1/x)/(x-1).

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We'll differentiate the given function with respect to x:

df/dx = d/dx {[3-(1/x)] / (x-1)}

df/dx = d/dx [3/(x-1)] - d/dx [1/x(x-1)]

d/dx [3/(x-1)] = [(x-1)*d/dx(3) - 3*d/dx(x-1)]/(x-1)^2

d/dx [3/(x-1)] = [0*(x-1) - 3*1]/(x-1)^2

d/dx [3/(x-1)] = - 3/(x-1)^2 (1)

We'll differentiate the term d/dx [1/x(x-1)]:

d/dx [1/x(x-1)] = d/dx [1/(x^2 - x)]

d/dx [1/(x^2 - x)] = [(x^2 - x)*d/dx(1) - 1*d/dx(x^2 - x)]/x^2*(x-1)^2

d/dx [1/(x^2 - x)] = -(2x-1)/x^2*(x-1)^2 (2)

df/dx = (1) - (2)

df/dx = - 3/(x-1)^2 + (2x-1)/x^2*(x-1)^2

df/dx = (2x - 1 - 3x^2)/x^2*(x-1)^2

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