Question

How do you find the second derivative of `ln |(x + 1)^2(x - 2)^4 / (2x + 9)|`?

Answer 1

Rewrite using properties of logarithms before differentiating.

Explanation:

We'll call the function `f`.

`f(x) = ln(((x+1)^2(x-2)^4)/(2x+9)) = 2ln(x+1)+4ln(x-2)-ln(2x+9)`

Use `d/dx(lnu) = 1/u (du)/dx`

`f'(x) = 2/(x+1)+4/(x-2)-2/(2x+9)`

Now use `d/dx(1/u) = -1/u^2 (du)/dx` to get

`f''(x) = -2/(x+1)^2 - 4/(x-2)^2 + 4/(2x+9)^2`