How do you find the derivative of #f(x)=ln(x^5(x-2)^3)#?

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Answer 1 · 2017-03-31T19:22:32

see below

Use the following Properties of Logarithm to expand the problem before taking derivatives.

That is,

#f(x)=ln (x^5 (x-2)^3) #

#=ln x^5 +ln (x-2)^3#

#=5 ln x+3 ln(x-2)#

#color(blue)(f'(x)=5*1/x + 3*1/(x-2)#

#color(blue)(f'(x)=5/x +3/(x-2)#

#color(blue)(f'(x)=(5(x-2) +3x)/(x(x-2))#

#color(blue)(f'(x)=(5x-10+3x)/(x(x-2))#

#color(blue)(f'(x)=(8x-10)/(x^2-2x))#