Answer
$${\text{ }}f''\left( x \right) = 54x + \frac{2}{{{x^3}}},\,\,\,\,\,f''\left( 1 \right) = 5{\text{6}},\,\,\,\,f''\left( { - 3} \right) = - \frac{{4376}}{{27}}$$
Work Step by Step
$$\eqalign{
& f\left( x \right) = 9{x^3} + \frac{1}{x} \cr
& {\text{write }}\frac{1}{x}{\text{ as }}{x^{ - 1}} \cr
& f\left( x \right) = 9{x^3} + {x^{ - 1}} \cr
& {\text{find the derivative of }}f\left( x \right) \cr
& {\text{ }}f'\left( x \right) = \frac{d}{{dx}}\left[ {9{x^3} + {x^{ - 1}}} \right] \cr
& {\text{ }}f'\left( x \right) = \frac{d}{{dx}}\left[ {9{x^3}} \right] + \frac{d}{{dx}}\left[ {{x^{ - 1}}} \right] \cr
& {\text{use power rule }}\frac{d}{{dx}}\left[ {{x^n}} \right] = n{x^{n - 1}} \cr
& {\text{ }}f'\left( x \right) = 9\left( {3{x^2}} \right) - \left( {{x^{ - 2}}} \right) \cr
& {\text{ }}f'\left( x \right) = 27{x^2} - {x^{ - 2}} \cr
& \cr
& {\text{find the derivative of }}f'\left( x \right) \cr
& f''\left( x \right) = \frac{d}{{dx}}\left[ {27{x^2} - {x^{ - 2}}} \right] \cr
& {\text{ }}f''\left( x \right) = \frac{d}{{dx}}\left[ {27{x^2}} \right] - \frac{d}{{dx}}\left[ {{x^{ - 2}}} \right] \cr
& {\text{use power rule}} \cr
& {\text{ }}f''\left( x \right) = 27\left( {2x} \right) + 2\left( {{x^{ - 3}}} \right) \cr
& {\text{ }}f''\left( x \right) = 54x + \frac{2}{{{x^3}}} \cr
& \cr
& {\text{find }}f''\left( 1 \right){\text{ and }}f''\left( { - 3} \right) \cr
& f''\left( 1 \right) = 54\left( 1 \right) + \frac{2}{{{{\left( 1 \right)}^3}}} \cr
& f''\left( 1 \right) = 54 + 2 = 56 \cr
& {\text{and}} \cr
& f''\left( { - 3} \right) = 54\left( { - 3} \right) + \frac{2}{{{{\left( { - 3} \right)}^3}}} \cr
& f''\left( { - 3} \right) = - 162 - \frac{2}{{27}} = - \frac{{4376}}{{27}} \cr
& \cr} $$
