derivative of (x^6+2/(8x^2))

解

\frac{d}{d x} (\frac{x ^{6} + 2}{8 x ^{2}})

\frac{x ^{6} - 1}{2 x ^{3}}

解答ステップ

\frac{d}{d x} (\frac{x ^{6} + 2}{8 x ^{2}})

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= \frac{1}{8} \frac{d}{d x} (\frac{x ^{6} + 2}{x ^{2}})

商の法則を適用:

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

= \frac{1}{8} \cdot \frac{\frac{d}{d x} ( x ^{6} + 2 ) x ^{2} - \frac{d}{d x} ( x ^{2} ) ( x ^{6} + 2 )}{( x ^{2} ) ^{2}}

\frac{d}{d x} (x^{6} + 2) = 6 x^{5}

\frac{d}{d x} (x^{2}) = 2 x

= \frac{1}{8} \cdot \frac{6 x ^{5} x ^{2} - 2 x ( x ^{6} + 2 )}{( x ^{2} ) ^{2}}

簡素化

\frac{1}{8} \cdot \frac{6 x ^{5} x ^{2} - 2 x ( x ^{6} + 2 )}{( x ^{2} ) ^{2}} : \frac{x ^{6} - 1}{2 x ^{3}}

= \frac{x ^{6} - 1}{2 x ^{3}}