Question

How do you find the second derivative of `ln(x^3)`?

Answer 1

`\qquad \qquad \qquad "second derivative" \ = \ - 3/x^2 \quad.`

Explanation:

`"We can start by rewriting the function, using Rules of"`
`"Logarithms, to prepare it for differentiation:"`

`\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad f(x) \ = \ ln( x^3 )`

`\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad f(x) \ = \ 3 ln( x ) \qquad \qquad \qquad \quad \ color{blue}{ "Power Rule for Logs" }`

`"Now differentiation is much easier !!"`

`\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad f'(x) \ = \ 3 [ \ ln( x ) \ ]'`

`\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad f'(x) \ = \ 3 cdot 1/x`

`"Continue, writing" \ f'(x) \ "to prepare it for differentiation:"`

`\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad f'(x) \ = \ 3 x^{-1}.`

`"Now differentiation is much easier -- no Quotient Rule needed !!"`

`\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad f''(x) \ = \ 3 [ x^{-1} ]'.`

`\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad f''(x) \ = \ 3 [ (-1) x^{-2} ] \ = \ -3 x^{-2} .`

`\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad f''(x) \ = \ -3 cdot 1/x^2 \ = \ - 3/x^2 .`

`\qquad \qquad :. \qquad \qquad \qquad \qquad \quad \ f''(x) \ = \ - 3/x^2 .`

`"This is our answer."`

`"Summarizing:"`

`\qquad \qquad \qquad \qquad f(x) \ = \ ln( x^3 ) \qquad rArr \qquad f''(x) \ = \ - 3/x^2 quad.`