How do you differentiate #f(x)=(x^3-3x^2-3x-6)/((x-6)(x-1)# using the quotient rule?

1 Answer
Nov 8, 2017

#(x^4 - 14x^3 +42x^2 -24x-60) / [ (x^2 - 7x + 6)^2 ] #

Explanation:

We are given

# F(x) = (x^3 - 3x^2 -3x-6)/((x^2-7x+6) )#

Quotient Rules states that

#D[ f(x)/g(x) ] = [ (g(x) f'(x) - f(x) g'(x)) / { {g(x)} ^ 2) ] color(green) [A]#

For our problem, f(x) is the Numerator and g(x) is the Denominator

Hence, f(x) = #(x^3 - 3x^2 -3x-6) color(red) ( Function 1)#

g(x) = #(x^2-7x+6) color(red) ( Function 2) #

Using #color(red) ( Function 1)#, we will find f'(x)

#f'(x) = (3x^2 - 6x - 3)#

Using #color(red) ( Function 2)#, we will find g'(x)

#g'(x) = (2x -7)#

Next, we will substitute these intermediate results in #color(green) [A]#

We obtain,

#( (x^2 - 7x + 6) ( 3x^2 - 6x - 3 ) - (x^3 -3x^2-3x-6)(2x - 7)) / (x ^2-7x+6)^2 #

From now on, we just need to simplify algebraically to get to the final answer.

#( 3x^2 - 6x - 3 ) / (x ^2-7x+6) - ((x^3 -3x^2-3x-6)(2x - 7)) / (x ^2-7x+6)^2 #

We can see that the common denominator is # (x ^2-7x+6)^2 #

So, we can simplify the expression further to obtain

#(x^4 - 14x^3 + 42x^2 - 24x - 60) / (x^2 - 7x + 6)^2#

This will be the required answer.

Any further simplification needs more calculation steps.