Question

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

Answer 1

`(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.