derivative of log_{2}((x^2/(x-1)))

解答

\frac{d}{d x} (lo g_{2} (\frac{x ^{2}}{x - 1}))

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

求解步骤

\frac{d}{d x} (lo g_{2} (\frac{x ^{2}}{x - 1}))

使用对数计算法则:

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

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

将常数提出:

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

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

使用连锁律:

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

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

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

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

化简

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

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

x \to + 3 lim (\frac{π + x}{4})

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