derivative of 1/(log_{4(x)})

解

\frac{d}{d x} (\frac{1}{lo g _{4} ( x )})

- \frac{1}{2 ln ( 2 ) x lo g _{4} ( x ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{1}{lo g _{4} ( x )})

指数の規則を適用する:

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

= \frac{d}{d x} (lo g_{4} (x)^{- 1})

連鎖法則を適用:

- \frac{1}{( lo g _{4} ( x ) ) ^{2}} \frac{d}{d x} (lo g_{4} (x))

= - \frac{1}{( lo g _{4} ( x ) ) ^{2}} \frac{d}{d x} (lo g_{4} (x))

\frac{d}{d x} (lo g_{4} (x)) = \frac{1}{2 ln ( 2 ) x}

= - \frac{1}{( lo g _{4} ( x ) ) ^{2}} \cdot \frac{1}{2 ln ( 2 ) x}

簡素化

- \frac{1}{lo g _{4} ( x ) ^{2}} \cdot \frac{1}{2 ln ( 2 ) x} : - \frac{1}{2 ln ( 2 ) x lo g _{4} ( x ) ^{2}}

= - \frac{1}{2 ln ( 2 ) x lo g _{4} ( x ) ^{2}}