Question

How can you take second derivative of `(x^2+12)/(x-2)` ?

I know that the first derivative is `{x^2-4x-12}/{(x-2)^2}` but I'm not sure how to take the second derivative. Please list detailed steps to take the second derivative, thanks!

Answer 1

`d^2/dx^2((x^2+12)/(x-2))=32/(x-2)^3`

Explanation:

Since you know that `d/dx((x^2+12)/(x-2))=(x^2-4x-12)/(x-2)^2`, you can take `d/dx((x^2-4x-12)/(x-2)^2)=d^2/dx^2((x^2+12)/(x-2))`.
Quotient rule: `y=f(x)/g(x)->dy/dx=(f'(x)g(x)-g'(x)f(x))/[g(x)]^2`
`:.d/dx((x^2-4x-12)/(x-2)^2)=((2x-4)(x-2)^2-2(x-2)(x^2-4x-12))/(x-2)^4`
`d/dx((x^2-4x-12)/(x-2)^2)=((2x-4)(x-2)^2-(2x-4)(x^2-4x-12))/(x-2)^4`
`d/dx((x^2-4x-12)/(x-2)^2)=((2x-4)[(x^2-4x+4)-(x^2-4x-12)])/(x-2)^4`
`d/dx((x^2-4x-12)/(x-2)^2)=((2x-4)(16))/(x-2)^4=(32(x-2))/(x-2)^4`
`:.d/dx((x^2-4x-12)/(x-2)^2)=32/(x-2)^3`