derivative of (ln(x+2)/(4x^{-2)})

解答

\frac{d}{d x} (\frac{ln ( x + 2 )}{4 x ^{- 2}})

\frac{x ( x + 2 ln ( x + 2 ) ( x + 2 ) )}{4 ( x + 2 )}

求解步骤

\frac{d}{d x} (\frac{ln ( x + 2 )}{4 x ^{- 2}})

将常数提出:

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

= \frac{1}{4} \frac{d}{d x} (\frac{ln ( x + 2 )}{x ^{- 2}})

使用微分除法定则:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{1}{4} \cdot \frac{\frac{d}{d x} ( ln ( x + 2 ) ) x ^{- 2} - \frac{d}{d x} ( x ^{- 2} ) ln ( x + 2 )}{( x ^{- 2} ) ^{2}}

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

\frac{d}{d x} (x^{- 2}) = - \frac{2}{x ^{3}}

= \frac{1}{4} \cdot \frac{\frac{1}{x + 2} x ^{- 2} - ( - \frac{2}{x ^{3}} ) ln ( x + 2 )}{( x ^{- 2} ) ^{2}}

化简

\frac{1}{4} \cdot \frac{\frac{1}{x + 2} x ^{- 2} - ( - \frac{2}{x ^{3}} ) ln ( x + 2 )}{( x ^{- 2} ) ^{2}} : \frac{x ( x + 2 ln ( x + 2 ) ( x + 2 ) )}{4 ( x + 2 )}

= \frac{x ( x + 2 ln ( x + 2 ) ( x + 2 ) )}{4 ( x + 2 )}

in v erse l a pl a ce \frac{2 s + 1}{s ^{3}}

d er i v a t i v e h (x) = \frac{x ^{2} + 4 x + 1}{3 x + 5}

\frac{d}{d x} (5 x^{2 x})

\int 2 x \cdot e^{2 x} d x

\int (1 + \frac{1}{u})^{- 3} (\frac{1}{u ^{2}}) d u