derivative (2x^9-5x+sqrt(x))/(x^4)

解

導関数

\frac{2 x ^{9} - 5 x + x}{x ^{4}}

\frac{20 x ^{9} + 30 x - 7 x}{2 x ^{5}}

解答ステップ

\frac{d}{d x} (\frac{2 x ^{9} - 5 x + x}{x ^{4}})

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d}{d x} ( 2 x ^{9} - 5 x + x ) x ^{4} - \frac{d}{d x} ( x ^{4} ) ( 2 x ^{9} - 5 x + x )}{( x ^{4} ) ^{2}}

\frac{d}{d x} (2 x^{9} - 5 x + x) = 18 x^{8} - 5 + \frac{1}{2 x}

\frac{d}{d x} (x^{4}) = 4 x^{3}

= \frac{( 18 x ^{8} - 5 + \frac{1}{2 x} ) x ^{4} - 4 x ^{3} ( 2 x ^{9} - 5 x + x )}{( x ^{4} ) ^{2}}

簡素化

\frac{( 18 x ^{8} - 5 + \frac{1}{2 x} ) x ^{4} - 4 x ^{3} ( 2 x ^{9} - 5 x + x )}{( x ^{4} ) ^{2}} : \frac{20 x ^{9} + 30 x - 7 x}{2 x ^{5}}

= \frac{20 x ^{9} + 30 x - 7 x}{2 x ^{5}}