How do you differentiate #f(x)=(x-5)/(x-3)^2# using the quotient rule?

1 Answer
Vikki
Dec 18, 2015

#f'(x)=(-x+7)/(x-3)^3 = -(x-7)/(x-3)^3#

Explanation:

How to differentiate using the quotient rule?

Given #f(x)= (x-5) /(x-3)^2#

#f'(x)= ((x-3)^2 d/dx(x-5) - (x-5)d/dx(x-3)^2)/((x-3)^2)^2#

#=((x-3)^2(1) -(x-5)(2(x-3)(1)))/(x-3)^4#

#=(color(red)((x-3))(x-3) -2color(red)((x-3))(x-5))/(color(red)((x-3))(x-3)^3)#

Factor out the greatest common factor #(x-3)#

#f'(x) = ((x-3)-2(x-5))/(x-3)^3#

#= (x-3-2x+10)/(x-3)^3#

#f'(x)=(-x+7)/(x-3)^3 = -(x-7)/(x-3)^3#

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