derivative of ({f}(x)/(x^5))

解

\frac{d}{d x} (\frac{f ( x )}{x ^{5}})

\frac{x \frac{d}{d x} ( f ( x ) ) - 5 f ( x )}{x ^{6}}

解答ステップ

\frac{d}{d x} (\frac{f ( x )}{x ^{5}})

商の法則を適用:

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

= \frac{\frac{d}{d x} ( f ( x ) ) x ^{5} - \frac{d}{d x} ( x ^{5} ) f ( x )}{( x ^{5} ) ^{2}}

\frac{d}{d x} (x^{5}) = 5 x^{4}

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

簡素化

\frac{\frac{d}{d x} ( f ( x ) ) x ^{5} - 5 x ^{4} f ( x )}{( x ^{5} ) ^{2}} : \frac{x \frac{d}{d x} ( f ( x ) ) - 5 f ( x )}{x ^{6}}

= \frac{x \frac{d}{d x} ( f ( x ) ) - 5 f ( x )}{x ^{6}}