Calculus of a Single Variable, Chapter 5, 5.5, Section 5.5, Problem 51

y=log_5 sqrt(x^2-1)
Before taking the derivative of the function, express the radical in exponent form.
y = log_5 (x^2-1)^(1/2)
Then, apply the logarithm rule log_b (a^m) = m* log_b(a) .
y = 1/2log_5 (x^2-1)
From here, proceed to take the derivative of the function. Take note that the derivative formula of logarithm is d/dx [log_b (u)] = 1/(ln(b) * u) * (du)/dx .
Applying this formula, the derivative of the function will be:
(dy)/dx = d/dx[1/2log_5 (x^2-1)]
(dy)/dx = 1/2 d/dx[log_5 (x^2-1)]
(dy)/dx = 1/2 *1/(ln(5)*(x^2-1)) * d/dx(x^2-1)
(dy)/dx = 1/2 *1/(ln(5)*(x^2-1)) *2x
(dy)/dx = (2x)/(2(x^2-1)ln(5))
(dy)/dx = x/((x^2-1)ln(5))

Therefore, the derivative of the function is (dy)/dx = x/((x^2-1)ln(5)) .