derivative of log_{4}(\sqrt[3]{x})

解

\frac{d}{d x} (lo g_{4} (3 x))

\frac{1}{6 ln ( 2 ) 3 x x ^{\frac{2}{3}}}

解答ステップ

\frac{d}{d x} (lo g_{4} (3 x))

対数の規則を適用する:

lo g_{a} (b) = \frac{ln ( b )}{ln ( a )}

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

定数を除去:

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

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

連鎖法則を適用:

\frac{1}{3 x} \frac{d}{d x} (3 x)

= \frac{1}{3 x} \frac{d}{d x} (3 x)

\frac{d}{d x} (3 x) = \frac{1}{3 x ^{\frac{2}{3}}}

= \frac{1}{ln ( 4 )} \cdot \frac{1}{3 x} \cdot \frac{1}{3 x ^{\frac{2}{3}}}

簡素化

\frac{1}{ln ( 4 )} \cdot \frac{1}{3 x} \cdot \frac{1}{3 x ^{\frac{2}{3}}} : \frac{1}{6 ln ( 2 ) 3 x x ^{\frac{2}{3}}}

= \frac{1}{6 ln ( 2 ) 3 x x ^{\frac{2}{3}}}