Question

What is the second derivative of `f(x)= ln (x^2+2x)`?

Answer 1

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

Explanation:

Use the substitution `t = x^2+2x`
Then `f(x) = lnt`

`(df)/dt = 1/t`
`(dt)/dx =2x +2`

`(df)/dx = (df)/dt * (dt)/dx = (1/(x^2+2x))* (2x+2)`
`f'(x) = (2(x+1))/(x^2+2x)`

We can now use the quotient rule to find the second derivative

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

`f''(x) = (2x^2 +4x -2x^2 - 8x -4)/(x^2+2)^2`
`f''(x) =-4(x+1)/(x^2+2)^2`