(x^8cos(3x)) /(6+3x^2) ^4 find dy/dx with logerthmic rule

Answer 1

The rule for logarithmic differentiation is

f' = (ln(f))'f

f = x⁸cos(3x)/(6+3x²)⁴

ln(f) = 8ln(x) + ln(cos(3x)) - 4ln(6+3x²)

(ln(f))' = 8/x - 3sin(3x)/cos(3x) - 24x/(6+3x²)

f' = (8/x - 3sin(3x)/cos(3x) - 24x/(6+3x²)) x⁸cos(3x)/(6+3x²)⁴