If ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}f(x)dx=\frac{2}{5}{x}^5-\frac{1}{3}{x}^{2}C,$ then determine a formula for $f(x).$

$($A$)\frac{2}{25}{x}^6-\frac{1}{6}{x}^{3}C$

$($D$)\frac{1}{15}{x}^6-\frac{1}{9}{x}^{3}C$

$($B$)2x^4-\frac{2}{3}x$

$($E$)\frac{1}{15}{x}^6-\frac{1}{9}{x}^3$

$($c$)\frac{8}{(5{)}^4}-\frac{1}{3}xc$

$($A$)A$

$($B$)B$