derivative of log_{ln(x}(e))

解

\frac{d}{d x} (lo g_{l n (x)} (e))

- \frac{1}{x ln ^{2} ( ln ( x ) ) ln ( x )}

解答ステップ

\frac{d}{d x} (lo g_{l n (x)} (e))

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 ( e )}{ln ( ln ( x ) )})

定数を除去:

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

= ln (e) \frac{d}{d x} (\frac{1}{ln ( ln ( x ) )})

商の法則を適用:

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

= ln (e) \frac{\frac{d}{d x} ( 1 ) ln ( ln ( x ) ) - \frac{d}{d x} ( ln ( ln ( x ) ) ) 1}{( ln ( ln ( x ) ) ) ^{2}}

\frac{d}{d x} (1) = 0

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

= ln (e) \frac{0 \cdot ln ( ln ( x ) ) - \frac{1}{x l n ( x )} \cdot 1}{( ln ( ln ( x ) ) ) ^{2}}

簡素化

ln (e) \frac{0 \cdot ln ( ln ( x ) ) - \frac{1}{x l n ( x )} \cdot 1}{ln ^{2} ( ln ( x ) )} : - \frac{1}{x ln ^{2} ( ln ( x ) ) ln ( x )}

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