Question

How do you differentiate `(x-4)/(x^2+2)`?

Answer 1

`f'(x)=(-x^2+8x+2)/(x-4)^2`

Explanation:

Apply the quotient rule which states:

`f(x)=g(x)/(h(x))->f'(x)=(g'(x)h(x)-h'(x)g(x))/((g(x))^2)`

Let

`g(x)=x-4`

`h(x)=x^2+2`

Thus,

`g'(x)=1`

`h'(x)=2x`

Now plugging into the formula:

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

Simplify:

`f'(x)=((x^2+2)-(2x^2-8x))/(x-4)^2`

`f'(x)=(-x^2+8x+2)/(x-4)^2`

Answer 2

`(8x-x^2+2)/(x^2+2)^2`

Explanation:

`"differentiate using the "color(blue)"quotient rule"`

`"given "y=(g(x))/(h(x))" then"`

`dy/dx=(h(x)g'(x)-g(x)h'(x))/(h(x))^2larrcolor(blue)"quotient rule"`

`g(x)=x-4rArrg'(x)=1`

`h(x)=x^2+2rArrh'(x)=2x`

`rArrd/dx((x-4)/(x^2+2))`

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

`=(8x-x^2+2)/(x^2+2)^2`