Question

What's the second derivative of `ln(f(x))`?

Answer 1

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

Explanation:

`y=ln(f(x))`

Through the chain rule and the knowledge that `d/dxln(x)=1/x`, we see that:

`dy/dx=1/f(x)*f'(x)=(f'(x))/f(x)`

To differentiate this again, we will use the quotient rule.

`(d^2y)/(dx^2)=(f(x)d/dxf'(x)-f'(x)d/dxf(x))/(f(x))^2`

`(d^2y)/(dx^2)=(f(x)f''(x)-(f'(x))^2)/(f(x))^2`