`f(x) = 8/((x - 2)^2)` Find the second derivative of the function.

Expert Answers
hkj1385 eNotes educator| Certified Educator

Note:- If y = (ax+b)^n ; where a,b,n are constants ; then dy/dx = na*x^(n-1)

Now, 

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

`or, y = 8*(x-2)^-2`

`or, y = -16(x-2)^-3`

`or, y = 48(x-2)^-4`

`or, y = 48/(x-2)^4`

``

loves2learn | Student

Using a quotient rule:

Given,

` `

`f(x)=a/b `

Then,

`f'(x)=((a')(b)-(a)(b'))/b^2 `

Therefore,

`f'(x)=((0)(x-2)^2-(8)(2(x-2)))/(x-2)^4 `

Simplify,

`f'(x)=-16/(x-2)^3 `

Take the derivative using a quotient rule again

`f''(x)=((0)(x-2)^3-(-16)(2)(x-2)^2(1))/(x-2)^6 `

Simplify this all down, and you are left with

`f''(x)=48/(x-2)^4 `