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

1 Answer
Mar 2, 2018

\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.