Answer
$$f'''\left( x \right) = 168x + 36,\,\,\,\,\,\,\,\,\,\,\,\,{f^{\left( 4 \right)}}\left( x \right) = 168$$
Work Step by Step
$$\eqalign{
& f\left( x \right) = 7{x^4} + 6{x^3} + 5{x^2} + 4x + 3 \cr
& {\text{find the derivative of }}f\left( x \right) \cr
& f'\left( x \right) = \frac{d}{{dx}}\left[ {7{x^4} + 6{x^3} + 5{x^2} + 4x + 3} \right] \cr
& {\text{apply the power rule }}\frac{d}{{dx}}\left[ {{x^n}} \right] = {x^{n - 1}}{\text{ to each term}} \cr
& f'\left( x \right) = 7\left( {4{x^3}} \right) + 6\left( {3{x^2}} \right) + 5\left( {2x} \right) + 4\left( 1 \right) + 0 \cr
& f'\left( x \right) = 28{x^3} + 18{x^2} + 10x + 4 \cr
& \cr
& {\text{find the derivative of }}f'\left( x \right) \cr
& f''\left( x \right) = \frac{d}{{dx}}\left[ {28{x^3} + 18{x^2} + 10x + 4} \right] \cr
& {\text{use the power rule}} \cr
& f''\left( x \right) = 28\left( {3{x^2}} \right) + 18\left( {2x} \right) + 10\left( 1 \right) + 0 \cr
& f''\left( x \right) = 84{x^2} + 36x + 10 \cr
& \cr
& {\text{find the derivative of }}f''\left( x \right) \cr
& f'''\left( x \right) = \frac{d}{{dx}}\left[ {84{x^2} + 36x + 10} \right] \cr
& {\text{use the power rule}} \cr
& f'''\left( x \right) = 84\left( {2x} \right) + 36\left( 1 \right) + 0 \cr
& f'''\left( x \right) = 168x + 36 \cr
& \cr
& {\text{find the derivative of }}f'''\left( x \right) \cr
& {f^{\left( 4 \right)}}\left( x \right) = \frac{d}{{dx}}\left[ {168x + 36} \right] \cr
& {\text{then}} \cr
& {f^{\left( 4 \right)}}\left( x \right) = 168 \cr} $$