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人気のある微分積分 >

derivative of log_{1/x}(e)

解

\frac{d}{d x} (lo g_{\frac{1}{x}} (e))

解

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

解答ステップ

\frac{d}{d x} (lo g_{\frac{1}{x}} (e))

簡素化

lo g_{\frac{1}{x}} (e) : - lo g_{x} (e)

= \frac{d}{d x} (- lo g_{x} (e))

定数を除去:

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

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

定数を除去:

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

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

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

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

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

簡素化

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

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

グラフ

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