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derivative y=log_{5}(x)+log_{5}(x^4)

解答

导数

y = lo g_{5} (x) + lo g_{5} (x^{4})

解答

\frac{5}{ln ( 5 ) x}

求解步骤

\frac{d}{d x} (lo g_{5} (x) + lo g_{5} (x^{4}))

化简

lo g_{5} (x) + lo g_{5} (x^{4}) : lo g_{5} (x^{5})

= \frac{d}{d x} (lo g_{5} (x^{5}))

使用对数计算法则:

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

= \frac{d}{d x} (\frac{ln ( x ^{5} )}{ln ( 5 )})

将常数提出:

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

= \frac{1}{ln ( 5 )} \frac{d}{d x} (ln (x^{5}))

使用连锁律:

\frac{1}{x ^{5}} \frac{d}{d x} (x^{5})

= \frac{1}{x ^{5}} \frac{d}{d x} (x^{5})

\frac{d}{d x} (x^{5}) = 5 x^{4}

= \frac{1}{ln ( 5 )} \cdot \frac{1}{x ^{5}} \cdot 5 x^{4}

化简

\frac{1}{ln ( 5 )} \cdot \frac{1}{x ^{5}} \cdot 5 x^{4} : \frac{5}{ln ( 5 ) x}

= \frac{5}{ln ( 5 ) x}

作图

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