derivative of log_{10}(sqrt(x))

解答

\frac{d}{d x} (lo g_{10} (x))

\frac{1}{2 ln ( 10 ) x}

求解步骤

\frac{d}{d x} (lo g_{10} (x))

使用对数计算法则:

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

= \frac{d}{d x} (\frac{ln ( x )}{ln ( 10 )})

将常数提出:

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

= \frac{1}{ln ( 10 )} \frac{d}{d x} (ln (x))

使用连锁律:

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

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

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

= \frac{1}{ln ( 10 )} \cdot \frac{1}{x} \cdot \frac{1}{2 x}

化简

\frac{1}{ln ( 10 )} \cdot \frac{1}{x} \cdot \frac{1}{2 x} : \frac{1}{2 ln ( 10 ) x}

= \frac{1}{2 ln ( 10 ) x}

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