Calculus of a Single Variable, Chapter 5, 5.5, Section 5.5, Problem 54

y=log_10((x^2-1)/x)
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
(dy)/dx = d/dx[log_10 ((x^2-1)/x)]
(dy)/dx =1/(ln (10) * ((x^2-1)/x)) * d/dx ((x^2-1)/x)
(dy)/dx = x/((x^2-1)ln(10) ) * d/dx((x^2-1)/x)
To get the derivative of (x^2-1)/x, apply quotient rule d/dx ((f(x))/(g(x))) = (g(x)*f'(x) - f(x)*g'(x))/[g(x)]^2 .
(dy)/dx = x/((x^2-1)ln(10) ) * (x*2x - (x^2-1)*1)/x^2
(dy)/dx = x/((x^2-1)ln(10)) * (2x^2 - x^2 + 1)/x^2
(dy)/(dx) = x/((x^2-1)ln(10)) * (x^2+1)/x^2
(dy)/dx = 1/((x^2-1)ln(10)) * (x^2+1)/x
(dy)/dx = (x^2+1)/(x(x^2-1)ln(10))

Therefore, the derivative of the function is (dy)/dx = (x^2+1)/(x(x^2-1)ln(10)) .