derivative of (2x^5+3x/(x^4))

解

\frac{d}{d x} (\frac{2 x ^{5} + 3 x}{x ^{4}})

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

解答ステップ

\frac{d}{d x} (\frac{2 x ^{5} + 3 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 ^{5} + 3 x ) x ^{4} - \frac{d}{d x} ( x ^{4} ) ( 2 x ^{5} + 3 x )}{( x ^{4} ) ^{2}}

\frac{d}{d x} (2 x^{5} + 3 x) = 10 x^{4} + 3

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

= \frac{( 10 x ^{4} + 3 ) x ^{4} - 4 x ^{3} ( 2 x ^{5} + 3 x )}{( x ^{4} ) ^{2}}

簡素化

\frac{( 10 x ^{4} + 3 ) x ^{4} - 4 x ^{3} ( 2 x ^{5} + 3 x )}{( x ^{4} ) ^{2}} : \frac{2 x ^{4} - 9}{x ^{4}}

= \frac{2 x ^{4} - 9}{x ^{4}}