Answer
$f'\left( x \right) = 50\left( {5{x^4} + 6x - 1} \right){\left( {{x^5} + 3{x^2} - x} \right)^{49}}$
Work Step by Step
$$\eqalign{
& f\left( x \right) = {\left( {{x^5} + 3{x^2} - x} \right)^{50}} \cr
& {\text{Differentiate}} \cr
& f'\left( x \right) = \frac{d}{{dx}}\left[ {{{\left( {{x^5} + 3{x^2} - x} \right)}^{50}}} \right] \cr
& {\text{Use The Power Rule Combined with the Chain Rule}} \cr
& \frac{d}{{dx}}{\left[ {g\left( x \right)} \right]^n} = n{\left[ {g\left( x \right)} \right]^{n - 1}}g'\left( x \right) \cr
& {\text{Therefore}}{\text{,}} \cr
& f'\left( x \right) = 50{\left( {{x^5} + 3{x^2} - x} \right)^{50 - 1}}\frac{d}{{dx}}\left[ {{x^5} + 3{x^2} - x} \right] \cr
& {\text{Computing derivatives}} \cr
& f'\left( x \right) = 50{\left( {{x^5} + 3{x^2} - x} \right)^{49}}\left( {5{x^4} + 6x - 1} \right) \cr
& f'\left( x \right) = 50\left( {5{x^4} + 6x - 1} \right){\left( {{x^5} + 3{x^2} - x} \right)^{49}} \cr} $$
