How do you differentiate $f (x) = \frac{x - 5}{{(x - 3)}^{2}}$ $f(x)=(x-5)/(x-3)^2$ using the quotient rule?

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Answer 1 · 2015-12-18T19:09:57

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

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 $(x-3)$

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

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

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

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