derivative of (x^{2/3}/(5+x+x^4))

解答

\frac{d}{d x} (\frac{x ^{\frac{2}{3}}}{5 + x + x ^{4}})

\frac{- 10 x ^{4} - x + 10}{3 x ^{\frac{1}{3}} ( 5 + x + x ^{4} ) ^{2}}

求解步骤

\frac{d}{d x} (\frac{x ^{\frac{2}{3}}}{5 + x + x ^{4}})

使用微分除法定则:

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

= \frac{\frac{d}{d x} ( x ^{\frac{2}{3}} ) ( 5 + x + x ^{4} ) - \frac{d}{d x} ( 5 + x + x ^{4} ) x ^{\frac{2}{3}}}{( 5 + x + x ^{4} ) ^{2}}

\frac{d}{d x} (x^{\frac{2}{3}}) = \frac{2}{3 x ^{\frac{1}{3}}}

\frac{d}{d x} (5 + x + x^{4}) = 4 x^{3} + 1

= \frac{\frac{2}{3 x ^{\frac{1}{3}}} ( 5 + x + x ^{4} ) - ( 4 x ^{3} + 1 ) x ^{\frac{2}{3}}}{( 5 + x + x ^{4} ) ^{2}}

化简

\frac{\frac{2}{3 x ^{\frac{1}{3}}} ( 5 + x + x ^{4} ) - ( 4 x ^{3} + 1 ) x ^{\frac{2}{3}}}{( 5 + x + x ^{4} ) ^{2}} : \frac{- 10 x ^{4} - x + 10}{3 x ^{\frac{1}{3}} ( 5 + x + x ^{4} ) ^{2}}

= \frac{- 10 x ^{4} - x + 10}{3 x ^{\frac{1}{3}} ( 5 + x + x ^{4} ) ^{2}}

\int 4 x + \frac{1}{4 x} d x

\int 5 x e^{- 2 x} d x

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

in v erse l a pl a ce \frac{s + 1}{( s ^{2} + 9 )}

t an g e n t f (x) = 2 x^{2} - 3 x - 2, at x = 6