derivative of 9/(4\sqrt[4]{x^3})

解

\frac{d}{d x} (\frac{9}{4 4 x ^{3}})

- \frac{27}{16 x ^{\frac{7}{4}}}

解答ステップ

\frac{d}{d x} (\frac{9}{4 4 x ^{3}})

定数を除去:

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

= \frac{9}{4} \frac{d}{d x} (\frac{1}{4 x ^{3}})

指数の規則を適用する:

\frac{1}{a} = a^{- 1}

= \frac{9}{4} \frac{d}{d x} ((x^{3})^{- \frac{1}{4}})

連鎖法則を適用:

- \frac{1}{4 ( x ^{3} ) ^{\frac{5}{4}}} \frac{d}{d x} (x^{3})

= - \frac{1}{4 ( x ^{3} ) ^{\frac{5}{4}}} \frac{d}{d x} (x^{3})

\frac{d}{d x} (x^{3}) = 3 x^{2}

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

簡素化

\frac{9}{4} (- \frac{1}{4 ( x ^{3} ) ^{\frac{5}{4}}} \cdot 3 x^{2}) : - \frac{27}{16 x ^{\frac{7}{4}}}

= - \frac{27}{16 x ^{\frac{7}{4}}}