How do you differentiate #f(x)= (x^2-5x+2)/ (x- 1 )# using the quotient rule?

Question

How do you differentiate #f(x)=  (x^2-5x+2)/  (x- 1   )# using the quotient rule?

Aurora Alvarado · Accepted Answer

#f'(x) = (x^2 - 2x + 3 )/(x - 1 )^2# using the #color(blue) " Quotient Rule "# #" If " f(x) = g(x)/(h(x)) rArr f'(x) = (h(x)g'(x) - g(x)h'(x))/[h(x) ]^2# hence# f'(x) = ((x-1) d/dx(x^2-5x+2) - (x^2-5x+2) d/dx(x-1))/(x-1)^2# #= ((x-1)(2x-5) - (x^2-5x+2).1)/(x-1)^2# #= (2x^2-5x-2x+5 -x^2+5x - 2 )/(x-1)^2 = (x^2 -2x + 3 )/(x-1)^2#

Adam Adkins · Answer

undefined To differentiate $ f(x) = \frac{x^2 - 5x + 2}{x - 1} $ using the quotient rule, follow these steps:

1. Identify $ u(x) $ and $ v(x) $.
   - Let $ u(x) = x^2 - 5x + 2 $ and $ v(x) = x - 1 $.

2. Calculate $ u'(x) $ and $ v'(x) $.
   - $ u'(x) = 2x - 5 $ and $ v'(x) = 1 $.

3. Apply the quotient rule:
   $$
   \frac{d}{dx}\left(\frac{u(x)}{v(x)}\right) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}
   $$

4. Substitute the values:
   $$
   f'(x) = \frac{(2x - 5)(x - 1) - (x^2 - 5x + 2)(1)}{(x - 1)^2}
   $$

5. Simplify the expression if necessary.