Logarithmic Differentiation

Question

Use logarithmic differentiation to find a formula for f '(x):

f(x) = ((x + 1)^(3/2) (x − 4)^(5/2)) / ((5x + 3)^(2/3))

Give clear steps and explanations

Answer 1

f(x) = (x+1)^3/2 (x−4)^5/2 / (5x+3)^2/3

(ln(f))(x) = (3/2)ln(x+1) + (5/2)ln(x-4) - (2/3)ln(5x+3)

(ln(f))'(x) = 3/(2(x+1)) + 5/(2(x-4)) - 10/3(5x+3)

The general formula of logarithmic differentiation is

f' = (ln(f))' × f =

(3/(2(x+1)) + 5/(2(x-4)) - 10/(3(5x+3))) (x+1)^3/2 (x−4)^5/2 / (5x+3)^2/3