derivative of (6x^7/(5tan(x)))

解

\frac{d}{d x} (\frac{6 x ^{7}}{5 tan ( x )})

\frac{6 ( 7 x ^{6} tan ( x ) - x ^{7} sec ^{2} ( x ) )}{5 tan ^{2} ( x )}

解答ステップ

\frac{d}{d x} (\frac{6 x ^{7}}{5 tan ( x )})

定数を除去:

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

= \frac{6}{5} \frac{d}{d x} (\frac{x ^{7}}{tan ( x )})

商の法則を適用:

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

= \frac{6}{5} \cdot \frac{\frac{d}{d x} ( x ^{7} ) tan ( x ) - \frac{d}{d x} ( tan ( x ) ) x ^{7}}{( tan ( x ) ) ^{2}}

\frac{d}{d x} (x^{7}) = 7 x^{6}

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

= \frac{6}{5} \cdot \frac{7 x ^{6} tan ( x ) - sec ^{2} ( x ) x ^{7}}{( tan ( x ) ) ^{2}}

分数を乗じる:

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

= \frac{6 ( 7 x ^{6} tan ( x ) - x ^{7} sec ^{2} ( x ) )}{5 tan ^{2} ( x )}