Question

How do you find the first and second derivative of `ln(x^2-4)`?

Answer 1

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

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

Explanation:

let `f(x)=ln(x^2-4)`

`color(orange)"Reminder " color(red)(bar(ul(|color(white)(a/a)color(black)(d/dx(lnx)=1/x)color(white)(a/a)|)))`

and more generally. `color(red)(bar(ul(|color(white)(a/a)color(black)(d/dx(ln(g(x)))=(g'(x))/g(x))color(white)(a/a)|)))`

`rArrf'(x)=(2x)/(x^2-4)`

To find the second derivative, use the `color(blue)"quotient rule"`

Given `f(x)=g(x)/(h(x))" then"`

`color(red)(bar(ul(|color(white)(a/a)color(black)(f'(x)=(h(x)g'(x)-g(x)h'(x))/(h(x))^2)color(white)(a/a)|)))......(A)`

here `g(x)=2xrArrg'(x)=2`

and `h(x)=x^2-4rArrh'(x)=2x`

substitute these values into (A)

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

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