Chapter 5 - Graphs and the Derivative - 5.3 Higher Derivatives, Concavity, and the Second Derivative Test - 5.3 Exercises - Page 283: 22

$$f'''\left( x \right) = - 6{x^{ - 4}},\,\,\,\,\,\,\,\,{f^{\left( 4 \right)}}\left( x \right) = \frac{{24}}{{{x^5}}}$$

$$\eqalign{ & f\left( x \right) = \frac{{x + 1}}{x} \cr & {\text{distributive property}} \cr & f\left( x \right) = \frac{x}{x} + \frac{1}{x} \cr & f\left( x \right) = 1 + \frac{1}{x} \cr & or \cr & f\left( x \right) = 1 + {x^{ - 1}} \cr & {\text{find the derivative of }}f\left( x \right) \cr & f'\left( x \right) = \frac{d}{{dx}}\left[ {1 + {x^{ - 1}}} \right] \cr & {\text{Then}}{\text{, by the power rule}} \cr & f'\left( x \right) = 0 - {x^{ - 2}} \cr & f'\left( x \right) = - {x^{ - 2}} \cr & \cr & {\text{find the derivative of }}f'\left( x \right) \cr & f''\left( x \right) = \frac{d}{{dx}}\left[ { - {x^{ - 2}}} \right] \cr & {\text{by using the power rule }}\frac{d}{{dx}}\left[ {{x^n}} \right] = n{x^{n - 1}} \cr & f''\left( x \right) = 2{x^{ - 3}} \cr & \cr & {\text{find the derivative of }}f''\left( x \right) \cr & f'''\left( x \right) = \frac{d}{{dx}}\left[ {2{x^{ - 3}}} \right] \cr & {\text{by using the power rule }}\frac{d}{{dx}}\left[ {{x^n}} \right] = n{x^{n - 1}} \cr & f'''\left( x \right) = 2\left( { - 3} \right){x^{ - 4}} \cr & f'''\left( x \right) = - 6{x^{ - 4}} \cr & \cr & {\text{find the derivative of }}f'''\left( x \right) \cr & {f^{\left( 4 \right)}}\left( x \right) = \frac{d}{{dx}}\left[ { - 6{x^{ - 4}}} \right] \cr & {\text{by using the power rule }}\frac{d}{{dx}}\left[ {{x^n}} \right] = n{x^{n - 1}} \cr & {f^{\left( 4 \right)}}\left( x \right) = - 6\left( { - 4} \right){x^{ - 5}} \cr & {f^{\left( 4 \right)}}\left( x \right) = 24{x^{ - 5}} \cr & or \cr & {f^{\left( 4 \right)}}\left( x \right) = \frac{{24}}{{{x^5}}} \cr} $$