Calculus of a Single Variable, Chapter 5, 5.5, Section 5.5, Problem 53

f(x) = log_2 (x^2/(x-1))
The derivative formula of a logarithm is
d/dx [ log_a (u)] = 1/(ln(a) * u) * (du)/dx
Applying this formula, the derivative of the function will be:
f'(x) = d/dx [ log_2 (x^2/(x-1))]
f'(x) = 1/(ln(2) * (x^2/(x-1))) * d/dx(x^2/(x-1))
f'(x) = (x-1)/(x^2ln(2)) * d/dx(x^2/(x-1))
To take the derivative of x^2/(x-1) , apply the quotient rule d/dx[(f(x))/(g(x))] = (g(x)*f'(x) - f(x)*g'(x))/[g(x)]^2 .
f'(x) = (x-1)/(x^2ln(2)) * ((x-1)*2x - x^2*1)/(x-1)^2
f'(x) = (x-1)/(x^2ln(2)) * (2x^2-2x - x^2)/(x-1)^2
f'(x) = (x-1)/(x^2ln(2)) * (x^2 - 2x)/(x-1)^2
f'(x) = (x-1)/(x^2ln(2)) * (x(x-2))/(x-1)^2
f'(x) = 1/(xln(2))*(x-2)/(x-1)
f'(x)=(x-2)/(x(x-1)ln(2))

Therefore, the derivative of the function is f'(x)=(x-2)/(x(x-1)ln(2)) .