Question

How do you use the quotient rule to differentiate `f(x)= (-4x)/ (x^2 - 1) ^2`?

Answer 1

`(12x^2 + 4)/(x^2 - 1)^3`

Explanation:

Using the `color(blue)" Quotient rule "`

If `f(x)=g(x)/(h(x)) " then " f'(x) =(h(x).g'(x)-g(x).h'(x))/(h(x)^2)`

and`color(red)" chain rule "`

`d/dx[f(g(x))] = f'(g(x)). g'(x)`
`"-------------------------------------------------------------------------"`

here g(x) = - 4x → g'(x) = - 4

and h(x) `=(x^2-1)^2 rArr h'(x) =2(x^2-1) d/dx(x^2-1)=4x(x^2-1)`

substitute these results into f'(x) above:

`f'(x) =((x^2-1)^2.(-4) - (-4x).4x(x^2-1))/(x^2-1)^4`

`=(-4(x^2-1)^2 + 16x^2(x^2-1))/(x^2-1)^4`

`=( (x^2-1) [ -4(x^2-1) + 16x^2))/(x^2-1)^4`

`=(cancel(x^2-1) (12x^2 + 4))/(cancel(x^2-1) (x^2-1)^3)=(12x^2+4)/(x^2-1)^3`