Chapter 5 - Graphs and the Derivative - 5.3 Higher Derivatives, Concavity, and the Second Derivative Test - 5.3 Exercises - Page 283: 19

$$f'''\left( x \right) = 300{x^2} - 72x + 12,\,\,\,\,\,\,\,\,\,\,\,\,{f^{\left( 4 \right)}}\left( x \right) = 600x - 72$$

$$\eqalign{ & f\left( x \right) = 5{x^5} - 3{x^4} + 2{x^3} + 7{x^2} + 4 \cr & {\text{find the derivative of }}f\left( x \right) \cr & f'\left( x \right) = \frac{d}{{dx}}\left[ {5{x^5} - 3{x^4} + 2{x^3} + 7{x^2} + 4} \right] \cr & {\text{apply the power rule }}\frac{d}{{dx}}\left[ {{x^n}} \right] = {x^{n - 1}}{\text{ to each term}} \cr & f'\left( x \right) = 5\left( {5{x^4}} \right) - 3\left( {4{x^3}} \right) + 2\left( {3{x^2}} \right) + 7\left( {2x} \right) + 0 \cr & f'\left( x \right) = 25{x^4} - 12{x^3} + 6{x^2} + 14x \cr & \cr & {\text{find the derivative of }}f'\left( x \right) \cr & f''\left( x \right) = \frac{d}{{dx}}\left[ {25{x^4} - 12{x^3} + 6{x^2} + 14x} \right] \cr & {\text{use the power rule}} \cr & f''\left( x \right) = 25\left( {4{x^3}} \right) - 12\left( {3{x^2}} \right) + 6\left( {2x} \right) + 14\left( 1 \right) \cr & f''\left( x \right) = 100{x^3} - 36{x^2} + 12x + 14 \cr & \cr & {\text{find the derivative of }}f''\left( x \right) \cr & f'''\left( x \right) = \frac{d}{{dx}}\left[ {100{x^3} - 36{x^2} + 12x + 14} \right] \cr & {\text{use the power rule}} \cr & f'''\left( x \right) = 100\left( {3{x^2}} \right) - 36\left( {2x} \right) + 12\left( 1 \right) + 0 \cr & f'''\left( x \right) = 300{x^2} - 72x + 12 \cr & \cr & {\text{find the derivative of }}f'''\left( x \right) \cr & {f^{\left( 4 \right)}}\left( x \right) = \frac{d}{{dx}}\left[ {300{x^2} - 72x + 12} \right] \cr & {\text{then}} \cr & {f^{\left( 4 \right)}}\left( x \right) = 300\left( {2x} \right) - 72\left( 1 \right) + 0 \cr & {f^{\left( 4 \right)}}\left( x \right) = 600x - 72 \cr} $$