derivative log_{10}((1+x)/n)

解答

导数

lo g_{10} (\frac{1 + x}{n})

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

求解步骤

\frac{d}{d x} (lo g_{10} (\frac{1 + x}{n}))

使用对数计算法则:

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

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

将常数提出:

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

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

使用连锁律:

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

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

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

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

化简

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

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

d er i v a t i v e \frac{x ^{2}}{a ^{2}}

d er i v a t i v e \frac{ln ( x ^{2} )}{2}

d er i v a t i v e x = 7 t^{2} - t + 4

\frac{d}{d x} (\frac{1}{( a \cdot cos ( x ) + b \cdot sin ( x ) )})

d er i v a t i v e (1 + x)^{5}