Calculus of a Single Variable, Chapter 5, 5.5, Section 5.5, Problem 56

First transform the given function using properties of logarithms. Also remember that xlt1.
g(x) = log_5(4) - log_5(x^2) - log_5((1-x)^(1/2)) =
= log_5(4) - 2 log_5(|x|) - 1/2 log_5(1-x).
Now it is easy to perform differentiation:
g'(x) = -2/ln(5)*1/x + 1/(2 ln(5))*1/(1-x).