Question

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

Answer 1

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

Explanation:

To find the first derivative, use the chain rule in conjunction with the knowledge that `d/dx[lnx]=1/x`. The rule for the natural logarithm is `d/dx[lnu]=(u')/u`. Here, `u=x^2+x`, so

`f'(x)=(d/dx[x^2+2])/(x^2+2)=(2x)/(x^2+2)`

To find the second derivative, use the quotient rule. This gives:

`f''(x)=((x^2+2)d/dx[2x]-2xd/dx[x^2+2])/(x^2+2)^2`

These derivatives can be found through the power rule.

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

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