derivative y=log_{x}(e)

解答

导数

y = lo g_{x} (e)

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

求解步骤

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

使用对数计算法则:

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

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

将常数提出:

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

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

使用微分除法定则:

(\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}}

化简

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 )}

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