derivative of (\sqrt[5]{x^2}/(x^2+1))

解答

\frac{d}{d x} (\frac{5 x ^{2}}{x ^{2} + 1})

\frac{2 ( x ^{2} + 1 ) - 10 x ^{\frac{8}{5}} 5 x ^{2}}{5 x ^{\frac{3}{5}} ( x ^{2} + 1 ) ^{2}}

求解步骤

\frac{d}{d x} (\frac{5 x ^{2}}{x ^{2} + 1})

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} ( 5 x ^{2} ) ( x ^{2} + 1 ) - \frac{d}{d x} ( x ^{2} + 1 ) 5 x ^{2}}{( x ^{2} + 1 ) ^{2}}

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

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

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

化简

\frac{\frac{2}{5 x ^{\frac{3}{5}}} ( x ^{2} + 1 ) - 2 x 5 x ^{2}}{( x ^{2} + 1 ) ^{2}} : \frac{2 ( x ^{2} + 1 ) - 10 x ^{\frac{8}{5}} 5 x ^{2}}{5 x ^{\frac{3}{5}} ( x ^{2} + 1 ) ^{2}}

= \frac{2 ( x ^{2} + 1 ) - 10 x ^{\frac{8}{5}} 5 x ^{2}}{5 x ^{\frac{3}{5}} ( x ^{2} + 1 ) ^{2}}

\int (15 x^{13}) d x

\int 9 x^{2} - 4 x + 3 d x

p \to 4 lim (p^{2} + p + 5)

d er i v a t i v e e^{2 x} cos (5 x)

d er i v a t i v e f (x) = \frac{6 x}{3 + x}